electronic impurity states and screening …...corresponding linear theory are appreciable in both...
TRANSCRIPT
ELECTRONIC IMPURITY STATES
AND
SCREENING IN METALS AND SMICONDUCTORS
by
John Bethell
Thesis submitted for the Degree of Doctor of Philosophy
of the University of London.
Department of Physics Imperial College
London SW7
Dece.nber, 1974
ABSTRACT
Problems concerned with the inter-relationship of
electronic screening and the formation of localized electronic
states as&oci:ited with impurities are discussed in two t:Tpes
of •physic. ycter The mathematical technique employed in
each type of :system involves the use .of a separable potential.
For free-electron-like metals, a self-consistent non-
linear scheme is presented for the calculation of screening
.charge densities associated with impurities. The scheme
takes explicit account of bound state formation. The in-
purity potential is split into two parts:. the separable form •
of the first part permits the summation of an important cart
of the perturbation series for the screening charge density,
whilst the second part is weak and is treated in linear
screening theory. Numerical calculations are carried out
using model impurity potentials of the Heine-Abarenkov form
and by employing a result of Singwi et at. to deecribe exchange __—
and correlation within the electron gas. Corrections to the
corresponding linear theory are appreciable in both the
screening charge density and the screened potential. Compared
with linear theory, an accumulation. of charge of a realistic
magnitude is found within the impurity core, whilst the total
screening charge within the impurity cell is decreased from
its value in linear theory.
Symmetry-induced gapless semiconductors are considered, and
numerical computations of the temperature-dependent static
interband dielectric function of such systems are presented.
A separable model of a screened impurity potential is
introduced. .r.1,J1 is used to calculate a density of status
-associated impurity, and hence to discuss the
possible f3= of resonant donor and acceptor states.
No well-defii c'Lop.or states exist for any realisable set
of physical Darameters, whilst acceptor binding. energies and -
resonant widths are strongly dependent on the details of the
band structure and the screening process and also on the
temperature.
- iv -
ACKNOWLEDGEENTS
I ,,,icknovcedge with thanks the receipt of a Science •
Research Council Studentship during the course of the won:
described in ths thesis. Special thanks are due to
Dr. Roy Jacobs and Dr. David Sherrington who shared- the
supervicion of my work and who respectively suggested the
investigations into metals and semicondotors. I should
also like to express my gratitude to the whole of the
Solid State Theory Group of the Physics Department, Imperial
College, who have contributed much good advice and moral
support.
o0o
Se non e verc, e molto ben trovato.
CONTENTS
Page No.
ABSTRACT ii
ACKNOWLEDGENETS iv
INTRODUCTIOr - Screened Potentials and Impurity
States 1
CHAPTER 1 SCREENING THEORY: AN INTRODUCTION 2
1.1 Linear Screening Theory 4
(a) Perturbative Methods
(b) Self-Consistent Field Methods 7
1.2 Non-Linear Screening
(a) Perturbative Methods 9
.(b) Non-perturbative Methods 11
1.3 Theories Non-Linear in the External
Potential 14
CHAPTER 2 IMPURITY SCREENING IN SIMPLE METALS 20
2.1 The Non--Linear Problem 20
2.2 The Separable Potential 21
2.3 Construction of Charge Densities 25
2.4 Discussion of Charge Densities 34
2.5 A Self-Consistent Scheme 42
CHAPTER 3 COMPUTATIONS 44
3.1 The Model System 44
3:2 The Numerical Scheme 48
3.3 Numerical Behaviour
Page No.
CHAPTER 4 RESULTS AND FURTHER DEVELOPMENTS 61
4.1 Screening Charge 61
4.2 Charge Transfer 67
4.3 Screened Potentials and Bound States 68
4.4 Diagrammatic Arguments 76
4.5 Other Potentials 80
4.6 Summary 83
CHAPTER 5 ZERO-GAP SEMICONDUCTORS 84
5.1 Introduction 84
5.2 The Band.Structure 86
5.3 Dielectric Behaviour 91
CHAPTER 6 IMPURITY STATES IN ZERO-GAP SEMICONDUC-
. TORS 100
6.1 Introduction 100
6.2 The Model Potential 103
6.3 The Density of States 105
6.4 Impurity Levels: Activation Energies and
Widths 112
6.4.1 Coulomb Potential 112
6.4.2 Zero-Gap Screening 115
6.4.3 Metallic Screening 118
6.5 Conclusions 124
APPENDIX 126
REFERENCES 127
- I -
INTRODtTCTION
SCREENED POT 'TALS AND IMPURITY STATES
The pro,i,,2m of a potential perturbing a gas . of
acting elctrons has been a central one in the developmen,z
of many-bo3y theory. An important concspL to emerge fro
the problem is that of a screened poienrial, arising froz-
the many-body renormalization of the-bare perterlDing poten-
tial. The general problem has been sub act to intense
(lN investigation and has a huge literature - ' • In tniz thesis,
a few specialised instances of the problem be discussed,
and the results of sophisticated general calculations per-
formed by others will be used whenever possible.
One situation where the details of the screening of a
potential are particularly important is where an impurity is
introduced substitutionally into a pure crystal, and the
impurity and host differ in valence. The screened potential
associated with the impurity may be strong enough to bind
valence electrons or holes into localized states. In a typical
semiconductor, such as Ge or Si, screening is rather ineffective
and the binding energies of the donor and acceptor states are
well-understood(2)
Furthermore, these states are close to
the Fermi energy and exert an appreciable influence on the
transport and optical properties of the semiconductor.
In a metal, on the other hand, the sereendg process is
more difficult to understand, because of the much higher
density of screening electroe,s. No general reliable scherle
- 2 -
exists for calculating the binding energies of localized
states, or in .lee- ler determining he existence of such states.
This diffice compounded by the experimental inaccess-
ibility of these states, as they may be far removed from the
Fermi level. :7evel:theless, this single-impurity problem
has far-reaching consequences in its extension to the elec-
tronic structure of concentrated metallic alloys where local-
ization of charge and charge transfer are controversial topics c3)
The first four chapters of this thesis are therefore devoted
to a better understanding of the screening of a strong attractive
impurity potential in a metal: particular attention is focussed
on possible formation of localized electronic states.
In chapters 5 and 6, semiconductors are considered,
but the discussion centres on the special case of zero-gap
semiconductors(4)
In these systems, the formation of donor
and acceptor states depends critically on the band structn - e
and on the distinctive screening behaviour, and the states are
only quasi-localized.
Thus, the inter-relationship of bound states and screen-
ing is a recurrent theme in this thesis. .Bound states are
treated throughout via singular t-matrices and separable
potentials are used to evaluate these t-matrices.
iv(2)
- 3 --
CHAPTER 1
SCREENING Introduct. ion
Since thy.;:, thesis makes wide use of results from
screening theory, is useful to begin by outlining some
of the techniques in screening theory which will be referred
to.
It has already been noted that this thesis is concerned
with the many-body problem of a potential perturbing a gas
of N interacting electrons, that is the problem of a system
described by the Hamiltonian
A/
= [ 171 t_9
4/
where Ti is the kinetic energy of the ith electron, Al
i is
the perturbing potential experienced by that electron, and
vij
is the potential of interaction between the ith and jth
electrons. (The units employed are described in the
Appendix.) The classical approach to this problem is to
transform the Hamiltonian into the form
II-j(:(0
IL • (1.2)
Where (n) ; i=1, ...N(n)) is a set of one-particle Hamii-
tonians describing non-interacting quasi-particles (quasi-
electrons,.plasmons etc.)(5)
For many purposeS, however,
the behaviour of collective excitations such as p1asm6ns is
// c,..})
011.- e. 4-e6)1‘ 1/
( /
where
not required, the effects of these modes being present in
. the quasi- .tron part of H. The simplified Harniltonian
/ (1.3)
may then be used. That is, the electrons respond to a sir,ple
one-body potential, the screened potential. Vs ,
In practice the canonical transformation to (1.2) is
not particularly useful and two other techniques have been
widely used to obtain Vs: perturbation theory and self-
consistent field methods. Ingredients from both techniques
are used in this thesis, and both techniques are outlined in
this chapter.
§1.1 Linear Screening Theory
The calculation of Vs is most simple when the applied
potential V(r,t) is weak and Vs is a simple linear response
to it. Vs
can then be described completely in terms of a
single response function, the dynamic dielectric function
(1.4)
The greater part of this thesis concerns situations where
V(r,t) is strong and (1.4) does not - apply, but it is a useful
point of departure.
(a) Pertu
inethods
Pert;.1 4ive 112tboas of describing screening rely on
extensive use 'eynman diagrams(6)
The Hamiltonian (1.1)
is written in second-quantized notation and causal Green's
functions C and Co set up for both this liamiltorcian and the
unperturbed Hamiltonian
Po (1.5
The diagrammatic perturbation series for G is written out and
from it is extracted those diagrams corresponding to
(see Fig. 1.1): that is those diagrams containing at least
one V, no external Go, and which carnet be split by cutting
a single Go
line.
Now in order for Vs to take the form (1.4), it is
necessary for all diagrams in which V appears more than once
to be negligible. (It is these diagrams which will interest
us in later chapters.) The remaining diagrams have a diver-
gent k = 0 Fourier component. The procedure adopted is to
now retain only the most divergent terms in each order of
perturbation theory. This approximation is equivalent to
the Random Phase Approximation (RPA) of Bohm and Pines(7)
The remaining terms, illustrated in Fig. 1.2, constitute an
infinite series whose exact sum may be evaluated, and which
is not divergent. Thus, the RPA dielectric function
ERp
A
(k'0 is obtained. At finite temperatures T'.; this takes
the form
V
(e)
6
Figure 1,1 tc=s in the perturbation seriev
sereeii.ed potential. •
The die.7.a
is used.
(a) (6) (c)
Figure 1.2 The R?A series for the screened potentiA1
>a(
4
di •
- 7
R;72- (-; k / 2 /
1_ v-74) , \0 (1.6)
Here Q is the volume of the system, c. is the energy of an
electron of momentum k, f(E) is the Fermi function
1 (1.7)
and k.B is Boltzmann's constant. x
o is the free-electron
susceptibility which Lindhard(8) has computed.
The screened potential calculated in this approximation
is only partially successful; although the problem of diver-
gent terms in perturbation theory has been circumvented, the
pair correlation function of the electrons can take negative
values. This is unphysical. The methods of overcoming
this difficulty will be described in §1.2, but first the
other approach to calculating the dielectric functioa will
be described for comparison.
(b) Self-Consistent Field Methods
Under this heading, we collect all theories of screen-
ing which do not involve summing a perturbation series.
They inevitably involve the decoupling of equations of motion
and, although Any given decoupling is in principle equivalent
to a particular partial sun of a perturbation series(9), this
is not always easy to do. The oriinal Self-Consistent Field
method(10)
however, which is in all respects a linear theory,
E
is exactly equivalent to the EPA.
The Self-Consistent Field (SCF) method begins with
the one-electIron screened. potential Vs(r,t) and the one-
'electron. Hamiltonian
• / / 5
/1 / t",) = (0
The equation of motion for the operator p of the one-electron
density matrix is then set up:
(1.8)
(1.9)
A plane-wave representation using the states jk> is used and
the density matrix written
`> c'k/ > 4- • , . 1.10)
Here <klp(°)Iki-q> is the density matrix of the unper-
turbed electron gas and <kip(1) ik+o> is the linear response
f of the density matrix to the potential <kiV
s ik+a›. The
equation of motion is now linearized retaining only these two
terms. The solution is now made self-consistent by calculat-
ing Vs from p(1) using Poisson's equation:
V 2( v 8772z
where V is the applied potential and n the charge density due
to o(1) •
This step constitutes a second linearization of
the problem: in practice Vs will have an exchange contribution
which is non-linear in n. The SCF method is thus a linearized
Hartree theory. The SCF dielectric function is readily
deduced and is the same as the dielectric function of the
RPA.
§1.2 Non,'Linear Screening
It is clear from the previous section that two types
of restriction apply to the simplest quantum mechanical
screening theories. First, these theories refer to weak
applied potentials V, which may be treated in linear-response
theory. Second, they neglect-exchauge and correlation
effects within the electron gas. Whilst the first restriction
is not severe, since a weak V may often be. chosen, the second
restriction is not satisfactory at metallic densities of the
electron gas. Considerable effort has been expended in
correcting this shortcoming via both perturbative and non-
perturbative methods.
(a) Perturbative Methods
One of the most notable sophistications of linear
screening theory was due to Hubbard(11)
He noted that a
certain class of diagram excluded from the RPA series was
simply related to the diagrams of the RPA series. He tera,:!d
pairs of diagrams related in this way "exchange conjugate"
diagrams because they represent direct and exchange components
of the electron-electron interaction: such diagrams are
generated from each other by crossing pairs of Co lines.
Thus, in. Fig. 1.1, (c) and (d) are exchange conjugate diar=s.
- 10 -
Hubbard was able to make an approximate sum of these diagrLm.s
and the effect on RPA was 'to replace xo as follos:
(1.12)
where
' ) = 2 2 4 4 2. z
and kF is the Fermi momentum. It should be noted, however,
that diagram (f) of Fig. 1.1 is still excluded..
Although this calculation may be improved, non-p2rtur-
bative treatments of the electron gas have been more success- .
ful. Recently, however, Fishlock and Pendry(12)
have shown
that the results of non-perturbative calculations can also be
obtained perturbatively, if at a greater expense of effort.
In their comparison with non-perturbative calculations of
g(r), the electron pair distribution function, they have only
been able to compute g(o). Nevertheless, this calculation
indicates a future for perturbative methods of treating the
electron gas, and indeed a similar technique to theirs is
employed in Chapter 2. The essence of the technique is to
split the short range part of the electron-electron inter-
action into two parts,
• bh --= + (1.13)
such that 2 is weak and is a separable, non-local
potential. In their computations they have used, for.
example,
(1.14)
Since is weak, 1 is chiefly responsible for the ex-
change and correlation effects which RPA neglects; the
separable form of (1.14) makes calculations of these effects
possible. In terms of diagrams, Fishlock and Pendry include,
in additiOn to the exchange terms of Hubbard, terms like (e)
of Fig. 1.1.
(b) Non-Perturbative Methods
Non-perturbative methods of treating the non-linearities
in the screening of a weak potential in an electron gas here
received extensive study, notably by Siogwi and others(13) .
The basic ideas of such methods will now be described.
As in the SCF method, we begin wich the equation of motion
of the density matrix (1.9). Now, however, we use the com-
ponents
(1.15) 6'
of the density matrix, {r.} being the positions of the elec-
trons, and employ the full, many-body Hamiltonian (1.1).
• Furthermore, no linearisation is carried out: instead, the
case where the external potential V v.=an isA.es is considered,
Then,
- 12 -
(116)
where f_.Aomentum operator for the iLb. elec!:ron. In
the final supulation of (1.16), the k = q term is treated
separately. If the k Q terms are omitted, the RPA or SCF
result will ultimately be recovered: these Lerms are now
included by making a static approximation. They are re-
written as
e,
(1.17)
and the sum over j is replaced by its ensemble average, a
static quantity
e >
(1„18)
(1.18) is iust the well known static structure factor- S(g-k)
of the electron gas(1) which is related to the dielectric
function by
7 ) )fs rho ,
Consequently, the equation of motion (1.16) bccomes identical
with the SCF equation of motion, with the replacement
(1.21)
t — •
- 13 -
Cr( , (1 2_(.7))
where
The dielectric function becomes
E(k 6d) /0 ri,/ /4
2.17 c. , 6(.1;
, 2 r-v / 77"/ ) ow) o ( //
(1.22)
and (1.19), (1.21) and (1.22) can be solved self-consistently
to obtain G(k) or e(k,u). Although this scheme gives good
results for large k, some modification is necessary for ,mall
k. Fortunately, the G(k) obtained in the two regimes of k
may be fitted most satisfactorily to the function
[I- ex12 f )/ rc•- r1 (1.23)
where the parameters A and B depend on the density of the
electron gas, This simple parameterization of electron
correlations will be employed in Chapter 3.
The self-consistent scheme which has just been des-
' A) cribed has been extended by Sj8lander and Stott to take
- 14 -
account of mixed systems, such as pc3itrons, protons and
impurities interacting with the electron gas,. They. accom-
plish this by the introduction of partial static structure
factors. Their approach is particularly successful for
describing positron annihilation rates in solids and yet
predicts unphysical electron densities around positively
charged impurities. They attribute this failure to the
. formation of bound states.
We remark that the static approximation (1.18) is
equivalent to neglecting the variation with electron energy
of electron-electron correlations. For the repulsive electron-
electron interaction this is a satisfactory procedure. How-
ever, when we introduce the attractive potential due to an
impurity, we shall see that the energy dependence of electron-
impurity correlations is vital: indeed, the t-matrix for
electron-impurity scattering is singular as a function of
energy if the impurity potential is strong enough for a bound
state to be formed. It is for this reason that in Chapter 2
we incorporate into our self-consistent scheme a _perturbative,
energy-dependent treatment of electron-impurity scattering:
in this way we hope to avoid the unphysical results of
Sj8lander and Stott.
§1.3 Theories Non-Linear in the External Potential
When a strong potential, for example an impurity poten-
tial, is screened by an electron gas, we have seen that a
theory which treats the potential in non-linear way is required.
- 15 -
However, before describing such theories, it is instructive
to see how such treatments'may be avoided in many circumstances.
When a crystalline array of ions is embedded in an
electron gas it is well known that the effective ionic.poten-
tials are weak and can be treated succc by linear
screening theory. This phenomenon may be explained by the
introduction of a pseudopotential(15) to replace the complicated
ionic potential. The pseudopotential is constructed so as to
represent the repulsive effect of the core electrons of the
ions in a simple way, which will describe the behaviour of
states outside the core, whilst giving no information about
the potential experienced by the core states themselves.
The most fundamental method of constructing such a
pseudopotential. is to project the nuclear potential on to a
set of states which are orthogonal to the states of the ionic
core. For many purposes, though, it is algebraically simpler
and more physically appealing to use to model potential of Heine
and Abarenkov(16)
In its simplest form it is
r i ( E) -2Z > ,„1 (1.24)
Here AL(E) is a constant depending on the energy E and on
• the orbital quantum numbers L = {Z,m) of the electronic state
on which it operates, Z is the total charge on the ion and
RIB is the model core radius. The constants A l (E) are deter-
- 16 -
mined by fitting to spectroscopic data the asymptotic form
as r ÷ c. of the wavefuncti.ons tPT (r,E) winch are solntions,
finite at the orl,gin, of
) /Z
(1.25)
where H0 is the kinetic energy operator of an electron.
In many situations, the repulsion represented by
AL(E) is appreciable and the pseudopotential is weak. In
pure free-electron-like metals, alkali metals for example,
the ionic pseudopotentials have little effect on the valence
electrons. In Chapter 3, the impurity system which we intro-.
duce has a free-electron-like metal as a host, and we ignore
scattering from the host ions in that situation. If a sub-
stitutional impurity is introduced into the crystal, the
pseudopotential on the impurity site is changed and the valence
electrons scatter from this change in potential. Now, if
the host and impurity ions are of similar size and have the
same valence, this scattering will be weak and linear screening
theory may be adequate.
However, if the impurity ion has a higher valence than
the host, it is conceivable that the change in potential will
be sufficiently strong for a bound state to be formed, even
if the potential is screened linearly. Linear screening
theory would•then be most inappropriate. Such a potential
can nevertheless be treated by what is essentially an extension
of the pseudopotential method: the auxiliary neutral atom
- 17 -
model. .16, which follows a suggesti.on by
Ziman(17)
a:' • pen exrloited Dagens(1 8')
the
scattering made electri_cally neutral by the intro-
duction of rbiirry distribution of elecLronic charge.
The resultil- --aK -)otential is treated a Unnr, self-
consistent way, thus avoiding the problem of screenig a
strong attractive potential. Unfortunately, this scheme
takes no account of the nature o the states in the screening
cloud, in particular whether they are localized states or not.
To gain a proper understanding of the screening process, the
full non-linear problem must be tackled directly and not
avoided.
To-clarify the ways in which a strong external potentiL.1
affects screening theory, we write down a schematic self-
consistent procedure by which a screened potential Vs and p
screening charge density ps determine each other.
V s 3
A
Calculation A has received most attention: the only
non-linearities associated with it arise from the electron-
electron interaction. The exchange and correlation effects
which these non-linearities represent are handled in a similar
way to the corresponding non-linearities in the presence of
only a weak perturbing potential.
Calculation. B, on the other hand, is one :itich has
(1.28)
- 18 -
almost inva, i:)1, - eFn. linearized. This linear respons,,,
theory is adequate when become strong enough for
a bound st,' formed: in such circumstances ps
will
vary discontinuously as V is varied. Such discontinuous
behaviour corner: arise merely from treating calculation A
in a non-linear way.
The non-linear calculations of type A have been des-
cribed by Dagens(18)
and by other workers(19)
lypically,
they involve including in Vs terms of the form
/7 /( 'Co -74. 2- J
r- -7 t. (6 0 (1.27)
where po is the unperturbed charge density and n' the induced
change in pc,. Various schemes(20) exist for choosing an
appropriate functionalPXC probably the simplest being due to
Kohn and Sham(21)
The non-linearities of calculation 13 have received
some treatment by March and Murray(22)
They have provided
explicit expressions to the Hartree screening charge produced
by a potential V to all finite orders in V.
S (zpig /2- - 7-7`
(1.29)
r2
- 19 -
The explicit ions (1.29) have recently been used by
Alfred(23)
to rin corrections to linear-response theory.
However, such procedure can only go a little way beyond
linear-resper.s.2 '113ory and can never include bound states
unless terms to all orders in V are included.
Finally, it must be mentioned that within the Fermi-
Thomas approximation an exact numerical solution to the full
non-linear impurity screening problem has been given by Murray
and March(24)
Their approach involves solving the differen-
tial equation
7-z vs
cortsx
for the screened potential Vs. This method of approaching
the problem is of a different character to the others which
have been described, and it is difficult to relate the results
of the method to those of other methods because of the imper-
spicuity of the Fermi-Thomas approximation.
- 20
CHAPTER 2
IMPURITY F:Y.7 SIMPLE METALS
§2.1 The 'Zon-Line..r Problem
In this and the - next two chapters, we shall be con--
cerned with the problem of a single substitutional impurity
in a free-electron-like metal. We have seen in Chapter 1
that this situation is well represented by a potential
embedded in an electron gas, and that linear screening theory
is inadequate when this impurity potential is strong and
attractive, because of possible bound state formation. We
therefore set up a self-consistent scheme of the form (1.26),
The calculation A is treated in a formal way for the present,
but calculation B is studied in detail using a non-linear
procedure based on perturbation theory. Certain approxi-
mations are needed to make this perturbative calculation,
but all orders of perturbation theory are included so as to
incorporate the possibility of bound states. To achieve
this, the impurity potential is split into two parts. The
first part has a separable form which permits the t-matrix
of this part to be evaluated easily. The remaining part is
weak, and linear screening theory (1st order perturbation
theory) is employed to treat it. In this way a charge
density is calculated, ignoring, for the moment, the electron-
electron interaction. This charge density contains a term
representing two electrons in a bound state, a term represent-
- 21 -
ing the cha sic y in linear screening theory and other
terms .which a a CX,f2Ed .as integrals over momenta. These
final terms in detail and put i'lto a form
suitable for numerical. evaluation. The scheme is finally
made self-cc::4_,'.tnt and the electron-electron interaction
is re-introduced by inclusion of calculation A.
We begin by introducing the separable potential which
is to be used.
52.2 The Separable Potential
We shall need to discuss two types of potential in
what follows: the bare impurity potential and the impurity
potential screened in •a self-consistent way. The bare
potential U takes the form
2 Z ee55
(2.1.)
as r 0. where excess is the excess charge on the impurity
ion. Charge neutrality implies that, when U is self-
consistently screened to give Vs, then, for sufficiently
large r, Vs is essentially zero. Now, it is quite possible
that if U is strong enough, both U and Vs are strong enough
to have bound electronic states. We mention both (2.1) and
the possibility of bound states because we wish to emphasise
how these two properties are retained and distinctly displayed
in the analysis which follows.
We consider an impurity potential 0; screened or un-screened, strong enough to bind at least one electron in an
- 22 -
s-state $> of eigenenergy E. Wrjting Ho as the flamil-
tonian of a free electron,; we have
(F/+ 7J ) /- .2) rj
We separate L/ into two terms, each of which
1 = + (2.3)
where the constituent parts are defined by
= u1 7, < / ) <:-V / cy,> (2.4a)
- v . (2.4b)
is of a particularly simple form which we shall
describe as separable. By this we mean that its matrix
element between any two states kii> and :42> can he written
as a simple product of a functional of itp/> and a functional
of 14/2›. The choice (2.4a) will enable as to sum an impor-
tant part of the perturbation series for the charge density
in §2.3.
Separations like (2.3), (2,4) have been studied by
Weinberg(5)
d and applied by Fishlozk an Pendry(12) (c.f.
(1.13-(1.14)) to the problem of electron correlations in the
unperturbed electron gas of a metal. The separation (2.3)
- is, however, more trans?arent than that used by Fish-lock end
Pendry, because the state 174, > used in the sel:rarat..io:% heri a
3
clear physics] interpretation via cquation (2.2) and is not
merely a naCiem,itical consLruction.
We -!(- ,lote some important properties of the separation
- (2 .3) .r st. 1 r Fi ji g
i ve s rise to the same hound state 111)>
as
( f_. / t) )-- \, I , _ i / —i 'v's.> o / -- 7 ".
2.5)
1 We may therefore consider L,'1 to be a pseudopotential cor-
responding to the potential 11 and appropriate to the energy
Eo and orbital angular momentum 9 = C. Naturally, any other'
elgenstates to which L1 g
ives rise do not correspond to the
eigenstates of the potential 7.1: in particular, this is true
of the states of positive energy. 'However, the important
?; i point is that the bound state property of is present ex-
clusively in the part and that this property can be easily
dealt with by virtue of the separable form (2.4a).
We now turn our attention to
As a corollary of
(2.5), we have
(2.6)
/ r Thus, for energies close to E'2
constitutes a weak poten-
tial. For positive energies 0; may not behave as a weak
(26) potential. This is anticipated by the analysis of Pendry
and confirmed by the numerical calculations in Chapter 3.
Nevertheless, in what follows, u2 is assumed to be weak
enough for its effects to be treated by linear-perturbaticn
- 24
theory. This will clearly not be a good assumption if 7
is strong enough to have bound states. Unfortunately, the
non-local nature of V prevents us from establishing
rigorously whether or not has bound states. Certainly,
if the range in real space of is sufficiently short, /I
has no bound state: putting
I (71 0 E (2( ) (2.7)
where Uo is positive and 6(r) is the three-dimensional Dirac
6-function, we obtain in the Schr8dinger representation
-7-2t-,)1;(7&-/) =0
(2.8)
A second feature ofV arises when L/ is the unscreened 2
potential of an impurity carrying an excess charge. Since
11P> has a finite range, I1 has a finite range, so that
becomes local at long range and retains the behaviour (2.7)
2Z excess
(2.9)
Thus, the charge on the impurity is manifested only in the
part V2 of the potential: each of the features of the
. potentials described in the first paragraph of this section
- are separately displayed in 1), and (4. We note that,
although 11 may be weak enough for linear perturbation 2
- 25 -
theory to he used, it is still responsible. for inducing, by
virtue of excess
), the total of Z units of screening
/ ti charge. only lead to a redistribution cf this
screening' r:
Finally, we indicate how stronger potentials, having
many bound states, may be separated to facilitate pertur-
bative calculations. Suppose that a potential has a set
f/ipn>; n=1,...N1 of bound states, orthogonal to each other,
then we may write: 4/-
n v- / 4.;/
7}73z,\2
- 4=7
(2.10)
The constituent potentials V have the property
- VIR1> 77 '17 4 I %1> L 0 --)11--5 V %
( 2.1.1)
2- -• ./
The potentials 1 are non-Hermitian, but this is not a serious difficulty because of the property (2.11) which en-
sures that all negative eigenenergies arising from the zit are real. It is now a straightforward matter to treat the
1)- n by infinite order perturbation and V/ by linear pertur-
bation theory. Unfortunately, the terms of the perturbation
- 26 -
series involving different UTihave to be included, and
this is not easy.
We shall see in later chapters that the effect of
applying non-linear perturbation theory to ("1 of (2.3) is
most significant for the greatest binding energyis iEu l.
We therefore conclude that the essential non-line ar behaviour
sriSing out of (2.10) is contained in the representing
the deepest bound state of
Consequently, the simpler
separation (2.3) should be adequate to describe the essential
ncin-linear behaviour, provided that the deepest bound state
is much deeper than any other bound states of the potential.
This approximation is a good one when the potential in
question is a Coulomb potential and we assume it to be satis-
factory for the potentials whicb we study.
§2.3 Construction of Charge Densities
In this section, we calculate the screening charge
density induced when an unscreened potential. U is introduced
into an electron gas. We suppose that U is sufficiently
attractive for a separation of the form (2.3) to be performed
and ignore interactions between the electrons for the presenL.
The system is then described by the one electron Hamiltonian
(2.12)
' We also define Green's operators by
-- 27 -
Next we wriLe ,n the perturbation series for G, and in
line with r_hz ]:eark_s q (2.6), •truncate the series
- by excludi_-_ all terms in which U. , appears to orders higher
than one.
(2.15)
Here, T1 is the t-matrix of U1 satisfying
7(5) (2-) (6:), (2.16)
and 6T contains those terms in which both U1 and U2 appear,
so that
(2.17)
Our next step will be to calculate the charge density
arising from (2.15). Because of (2.9), the third term of
(2.15) must lead to the entire total screening charge, the
second and fourth terms representing redistribution of this
charge. We now compare these two redistribution terms, and
argue that toe term in TI will dominate over the term in 6T,
provided that U2 is much weaker than
We have seer. (2.8)
that U2 is weak when U is short-ranged, and consequently the
term in T1 will dominate most strongly when U is a screened
potential. Since we shall calculate charge dcnsities induced
by a self-consistently screened potential, we TioAe the r27pror.i-
mation of dropping the 67 ten.: from (2.5). Combinir, r,
- 28
and 6T, we tE:rrJs of (2.15) of the form
(-40 C_TO (2.18)
Thus, the rfc,..L: of the 6T term is to mo,'
the flux:of
electrons scattered by T1 from that represented by Go to
that represented by Go+Go U2 Go. The approximation of
dropping the term in ST is therefore most satisfactory when
the density of the electron gas is greatest and Go U2
introduces only a small fractional change in the flux.
The approximation is also necessitated by the complexity of
the terms in ST: whilst the charge density associated with
these terms may be written down formally as c'asily as that
associated with T1, numerical analysis of such terms would
be prohibitively difficult. In summary, our approximation
of neglecting ST is most satisfactory for
(a) smallexess
(b) short screening lengths
(c) small r
We now utilise the separable form (2.4a) of U1
equation (2.16) to obtain
&/t17> 7'77k/ < /&/1/-> /<V/6(///2
[<wv,/ ;7 <W(/0(//,/?7/>7 .(2.19)
- 29 -
The identity
) ' 0// /"."
(2.n)
which follows from (2.2), together with the definition (2.14
enables (2.19) to be written
)<-1Heef--0/v/) (2.21) .
Because of the form (2.4a) of Up the singularity at E = Ec
is the only singularity of T1(E) and we can proceed to cal-
culate the charge -density associated with the Green's function
(2.15).
We call the charge density of our system p(r) and
define its Fourier transform p(a) by
cq) (2.22) .
We use this definition, along with the well-known expression
for the charge density p(r,E) of electrons of energy E
\
—
(2.23)
to obtain an expression for p(a) which includes all occupied
states at zero temperature:
- 30 -
6 7 ,/;77? • 6 „-.7 (2.24)
-op
We recall.)...Jtation.,of Chapter 1 which is used here;
• 4/ / / = (2.23)
is the elgenvalue equation for free-electrons, and lk> is
normalized according to
(2.26)
We now write
= (2.27)
in which po, p1 and p2 derive respectively from the first
three terms of (2.15). We consider the three terms of
(2.27) separately.
po
is simply the charge density of the unperturbed
electron gas,
2 / 77--
12_ If°
(2.2S)
- 31 -
Ile-re N is Cc ., 1,7il. - of occupied k-states in the volume
f(E) is the cxion in the zero temperature limit,
and Sk kl Thre-dimensional Kronecker delta
r 4
( (2..29)
The factor of 2 arises, of course, from spin degeneracy
po(1) therefore has only a a = 0 component, and will be
written simply as p c) from now on.
We now turn our attention to the multiple-scattering
term (a) :
= 6-to dE 7,7E-is) //2-7Y> 1-4 .
(2.30)
(2.2), (2.25) and (2.14) are now used to establish the
identities
<'4/1 2.31)
i<14. - - (2.32)
and the real functions 15r(E),(E), Dr(E) and D (E) are
defined by
- 32 -
O s -cr* • • • 9- -6 (2.33)
,/ ••=-.7c . .1
(2. 14 )
(2.31), (2.32), (2.34) enable us to write
(E-z.g)/h_ip = I
/ — ) — ( 0
(2.35) '
Furthermore, since <flr> is spherically symmetric., <thlk> is
a real function of k only. Thus, the real and imaginary
parts of (2.35) are easily separated. We insert (2,35)
into (2.30), pick out the imaginary part using the identity
(2.36)
and perform the integral over energy. The 6-function t8i2ms
lead to Fermi functions, and we obtain
(1 ) gc) I7/>
Dt (1-- f(1.7.--0 4
;-1
- 33 -
) -1, j - (,, __ J17) ,---
ter/ ?40 (--o (
- r -0/ k
JJt
(2.37)
Finally we turn to p2(q). We recall the definition
(2.4b) of U2, and write p2 as the difference in charge
densities induced by U and U1 in 1st order perturbation
theory.
(7) = /,'
• (2.38)
p2(E) is the charge density associated with U and in RFA
for example,
21( =xD (.) i//iV (2.39)
where x0(q) the static limit of the function defined in
(1.6) and U(q) is the Fourier transform of U(r)n defined as
in (2.22).
p2"(1) is the screening charge density associated with
the non-local potential U1 in linear screening theory. The
screening of a non--local potential has been studied in detail
by Animalu(27)
, but here a completely analogous result 'Lc his
is obtained by the same type of analysis as led to (2.37).
Eh
- 34 -
0 / <7A r rY/,=\
(2.40)
52..4 Discussion of Charge Densities
We now study the expressions (2.37) and (2.40) for
p i(a) and n2"(a): we shall first obtain a physical under-
standing of them and then cast them Into a form which will
permit them to be evaluated numerically.
We begin by examining the general form of B(E)
defined in (2.32). (2.33) and (2.36) .,,nable us to write
Consequently, Di(E) is non-zero only for energies E > - 0.
Furthermore, (2.41) approaches zero for large E by virtue of
the finite k-space range of <$1k>.r(E) may be written in
terms of D(E) in the dispersion relation
/:-L
(2.42)
It follows that 5(E) is real and positive for E < 0. The
- 35 -
vrieralform,;offl(E)andp.(E) are sketched in Fig. 2.1.
In particula wa may obL Ln directly Irflim (2,v2)
using the fact that <4) k> is normalized.
We are now in a position to understand the expression
(2.37) for pl(f1). Since f(E0) = 1, the first term in the
- early braces reduces to
r 1177-> /, e 0 (/-) 2
?Zo (-/L) < / 2 (2.44)
It consequently represents a pair of electrons (with opposite
spins) bound in the state 0 . The remaining terms of
(2.37) all arise from states in the band region (0 E
This is because the second and third terms both originate
from poles of the form (E - eki
, whilst the fourth terra
contains the factor D(E) which vanishes for E < 0. Further-
more, it is easy to establish that the sum of the second,
third and fourth terms of (2.37) vanishes in the limit F 0
We may obtain some idea of the effect of the second
and third terms of (2.37) by writing them in the q 0 limit:
,z _ /. en; / /7-2L- \ 1
6.4%,Y2 / / /eF,/ // (2.45)
It is well known that the q -- 0 limit of xo(q) is negative.
- 36 -
F
/ -1
Figure 2.1 The function,' D r E) D(E),
- 37 -
and it turns. o' It that the numerical values of P r (EF ) in
the system :s studied are Tositive. Thus, these
terms ;i? :,,:ati-ve contribution to o,(0), i.. a they
represent a J..:11sion of electronic charge. This repulsive
effect of the :ttraccive potential U1 is a significant
physical effect which linear perturbation theory does not
anticipate. We also remark that the limit (2.45) affords
a useful check on numerical computations. No such easy
check exists for the final term of (2.37), but, in practice,
it also represents charge repulsion.
It is not yet clear how (2.37) may be evaluated in
practice for non-zero q. In particular, the various poles
of the integrand prevent direct numerical integration. How-
ever, (2.37) may be put into a form in which direct numerical
integration is possible. To show this, we begin by writing
the second and third terms as
"F 1-1
c/L (1)/( — I
(2 .1: 6)
where
de, 9 _2
I _I —
el (2.47)
and p is the cosine of the angle between k and a. This is
possible because the summation in (2.17) is over all k and
because <lidk> and ek are functions of ilk only. Our aiD
- 38 -
is LoW to write (2.4b) as a sum of iategrals, each of which
is finite and can either be evaluated analytically or has
no poles in its integrand. If this is possible, then
(2.46) c‘an be evaluated by a simple numerical technique.
We consider the identity
fit (x-x-0)
to< (0).13("z----->,-G) dz. 0,/ )sir (x____;(0)ezr (2.48)
If we suppose that
(a) 8(x-x o' ) has a pole of some sort at x = xo
;
(b) 8(x-x0)dx can be performed analytically and i
is finite; and
(c) (a(x) - ct(xo))8(x-x
o) is non-singular,
then (2.48) constitutes a method of writing an integral with
a- singular integrand as a sum of integrals which can either
be evaluated analytically or by direct numerical integration,
A transformation like (2.48) is just what we require
to re-write (2.46), and, after several transformations of
this type, we obtain
r r% 7 2 jO Clif/6 J_10/)
, /7}
7/2k
(27
- 39 -
r-
v Q I 4 Z! F / 1
I C/ 2/'F/ V 02/4
(2.49)
The final term of (2.37) is a little harder to deal
with. We split the integrand into partial fractions and
introduce the function
(2.50)
which enables us to write
p EF
E- 1-7 .616 -WE) (-E-E0 a 7-
c(2.51)
Now, S(s) has the important properties
02) = 0 (2.52)
(2.53)
for E close to EF. Thus the fourth term of (2.37) splits
up into three terms, one with a non-singular integrand, and
the others with a simple pole and a logarithmic singularity
`A
// 1 e..1. 1? Le h( 31— 4" 1,/e-,4 - • f r *-, .
- 40 -
in each The additional logarithmic singularity
complicates 7rocedure of removing singularities, and it
has not pro -.D"le to obtain an entirely satisfactory
expression. 7theless, it is possible to write the
fourth term cs (2.37) as a sum of integrals, each of which
is finite, apa. t from two integrals which each have a
logarithmic singularity at q = 21:F. Since the original
expression had no such singularity, the two singular com-
ponents must cancel. It turns out that, in the numerical
calculations, satisfactory cancellation is obtained overall
but a very narrow range of q. This restriction is not
severe for present purposes, but if one wished to compute
Friedel-like oscillations in the charge density within this
scheme, the details of the behaviour close to q = 2kF would
have to be computed more precisely.
We eventually obtain for the fourth term of (2.37),
by applying identities like (2.48), and by using (2.52),
,- i/e 1 < tc- 74g> —0;
//() /7(k /%
P
— 41 —
7,4 vr -7 r 67_ zkr i
•
2 1)
16-1 , , 773 , 0
/ 2 • ) /a j ( i74 )
ji
/ Vr. / 7 /
X
/
/ 2 17Z
I/27424F (2 54) ---- $ /in ( C'/ k - 16` fr L
Here,
and we have introduced the step function
0(k- biz
The expression (2.40) for o2 " is ' susceptible to the
same treatment as the s e cond and third terms of (2.37). T n
I 2 ,
1
0
//.2
61/2_
particular, there is a simple checis on the q = 0 limit.
- 42 -
For arbitrary q, we obtain an expression identical to (2.49)
with the re. l.,3.cemcnt
(Wi (2.55)
<R
k 62 <WL 11'7 '//
52.5 A Self-Consistent Scheme
Now that we have calculated the charge density induced
by U, we are in a position to introduce the electron--electron
interaction 1J(q) and make the solution self-consistent.
We write the. total screened potential Vs, corresponding to
bare potential U, as follows:
vs( f ) -(/( f ) -,477(cp ;(
(2.56)
The second term on the right hand side of (2.36) is just the
Hartree potential of the screening charge; and the third
term F(p) is a non-linear functional of the screening charge
density reprasenting the exchange potential of the screening
charge. This potential his becomes self--consistent if the
charge densities are calculated in the field rather
than Then, using (2.39) may obtain
- 43 -
S(6,2) = r-
Schematically,
„ ) „fz( ? _ -e J
(2.57)
= y si
(2.58)
where y is .a non-linear functional of V s. Provided that
the correction y(Vs) to V S is small, then an iterative
solution to (2.58) is a strai7htforward proposition. A
disadvantage of (2.58) is that the non-linear effects of
the electron-electron and electron-impurity interactions
are inseparable. When we go on to study numerically a
scheme like (2.58), we shall use an approximation which
distinguishes these two non-linear effects more clearly.
- 44 -
CHAPTER 3
COMPUTATIONS
In this chapter, we describe how the self-consistent
scheme of Chapter 2 may be applied to a particular model
impurity potential and a particular model of a metallic
electron gas. The necessary numerical analysis is indicated
and the general numerical behaviour of the scheme is presented.
§3.1 The Model System
We consider a free-electron-like metal of valence Zc
into which a single impurity ion of valence Zi has been intro-
duced substitutionally. We, ignore deformation of the lattice
and scattering from the ions of the host, and represent the
situation by a single one-electron potential U embedded in
an electron gas of suitable density. U takes the form
(Z( a 6 /e.i6a
(3.1)
where U1 is the one-electron potential due to the isolated
impurity ion and U° the potential due to an isolated ion of
the host. We use for U1 and U° model potentials of the
Heine-Abarenkov form (1.24) appropriate to the deepest bound
s-states of the free ions. Whilst this choice is not parti-
cularly suitable for discussing band electrons, it yields,
for Z° -.Zo
1 and alkali metal or alkaline earth hosts, a
potential U with a bound state. Thus, this choice yields a
(3.2)
A, 2 Z. -7
- 45 -
potential U a f,?ature which is essential to our scheme.
The details (1c)re rc:inn of the potential are not
important. the fact that bound states are possible,
because it th, affect of valence rather than core electrons
on impurity :2=cening which we wish to discuss. U therefore
has the form
Ao
is a constant for any particular system. and we choose to
study alloy systems compounded. from elements of the same row
of the periodic table so that the same model core radius
- is suitable for both U° and
We now turn to the electron gas. Since we are chiefly
interested in the non-linear effects of the electron-impurity
interaction, we satisfy ourselves with a linearized treatment
of exchange and correlation in the electron gas; i-.e. a
treatment where the screened potential depends only linearly
on the density of screening charge. Such a linearized treat-
ment has been studied extensively by Singwi et al'.(13)'
who
find that correlation and exchange may be introduced into a
Hartree screening theory by the replacement (1.20). We
assume that this treatment, employing the function G(k) (1.23)
gives the essential ingredients of correlation and exchange
within our model, even though U is not a weak potential and
some of the electrons with which we are concerned are in
localized rather than free states.
- 46 -
Instead of (2.57), we have the self-consistent schema
V s(V -(A(672c
(3.3)
where • ) Y f IV( o (3.4)
((3.4) is not the dielectric function in the terminology of
Singwi et al.)
Schematically,
(3.5)
- where f3 represents a non-linear functional. (3.5) may be
solved for Vs by determining the limit V. = Vs of the sequence
V1, V2 .... of functions generated from the trial function Vo.
by
/n (3,6)
Vo must be chosen to obtain rapid convergence of the iterative
scheme (3.6).
(3.7)
proves to be a most satisfactory choice, but
(3.8)
also gives rapid convergence.
(3.11)
- 47 -
It co_ to define some of the quantities which
we shall calcyl thin this model. We define a charge
density the corresponding screening theory,
which is linaLr in the electron-impurity interaction, by
) (3.9)
.The correction to pZin
in our non-linear scheme is then
(3.10)
and the potential to which (3.10) gives rise is
Thus
175(f) = ) -77)«.A(
I f /' PJA (3.12)
We also examine the q 0 limits of these charge densities:
these represent the total screening charge present in the
system. Nor, since U(q) represents a charged object,
e:e7 ) —CZ —z°)-1)-(x). (3.13)
Consequently,
, (3.14)
- 48 -
On the other hand, the finite range of Vs implies that
4-7--,3> /-1 / ) = constant L
(3 .15 )
so that
/32 (3.16)
Thus, the correction pn2,(q) explicitly preserves overall
charge neutrality: it represents a redistribution of the
-o electrons represented by p,. (o).
kin
53.2 53.2 The Numerical Scheme
We now describe some of the numerical techniques which
have been employed to carry out the self-consistent calculations
described in Chapter 2 and 53.1. A block diagram showing
the more important stages in the computational scheme appears
in Fig. 3.1.
We begin by remarking on some of the general features
of the scheme. All of the functions which it uses (by virtue
of the spherical symmetry of rhe prcblem) are essentially
functions of a single variable. Corsecuenrly, no storage
problems arise OP the computer. Most ef the functions are
smooth: they may be tabulated at suitable in_e.rvals and
linear interpolation may be used to obtain internediate %31 :es,
No original techniques are involved in the scheme: the prin-
cipal difficulties arisc from the complexity with s
techniques are assembled. We shall therefore outline the
- 49 -
simple techniques and then indicate some of the problems
involved in assembling them.
The only numerical procedure, apart fro integration
. and interpolation, which has been used, is the solution of
the radiA Schrdinger equation tc obtain <r and E
The technique which has been used, has been adapted from a
method which has been widely employed for calculating loga-
rithmic derivatives for use in the augmented plane wave
method(28) An energy is chosen, and, with this energy, the
Schr8dinger equation is integrated radically outwards on a
logarithmic mesh, using first a Runge-Kutta and then a Milne
method, for about two interatomic distances. This calculation
is repeated at different energies, and an energy Eo, at which
the wavefunction <rk> is that of a bound state, is located.
Now, in general, the .analytic form of the wavefunction
as r a, for a finite ranged potential is
(E) E(5) , IL
Where K2
= -E. A(E) and B(E) are functions of energy only,
and at a bound state energy, E = Eo
2(Lf -o) -----
(3.18)
In practice, we are certain not to choose an energy exactly
equal to a bound state energy: B(E) remains finite and we
obtain the wavefunction
) (1.19';
- 50 -
Figure 5-1 2lock ' rroeed=e.
ParazIraters aad tiona
1`51=erical ocess
( FT = Fourier ,r: ns orm)
Decision
Details of "II:TEGRATIOff" Block (s next page):
clia-79ect ?
- 51 --
Bo:f t!?
( Pot; ena.t.
- -1—....„-r-4/ r i
IConstructnaxc-
V )
a;ive Schraciinacr
Normalize, inte3rate CI 1 ;x etemei
(k)
/et NTEGR ,c17 I 014
11]
G(k)
FT ftjt (r))
r"
(Ern 36:s
, B
30.5
F jui a 3i Lau
- 52 -
which diver 2:7- for large r. Nevertheless, provided E is
sufficiently to a bound state energy, there exists a
Kr' range. of r eve A(E)e /r is small by virtue of the
exponential r-,,d =')J' / r is small by virtue of B(E) being
small. ip(r„) __, flat and small in this region, and the
existence of such a flat region in the numerically integrated
wavefunction is taken as evidence that E is close to Eo. In
practice, it is found that once such a flat region has been
obtained, further improvement of the chosen value of E leads
to negligible change in the short range part of gr,E).
In a number of the simple integrals which have been
performed (including Fourier transforms), the integrands have
an infinite range. The range of integration is divided into
two parts in those cases: Simpson's Rule or the trapezium
rule is used for the short range part of the integral, and
the asymptotic form of the integrand at long range enables
an approximate long range contribution to be evaluated
analytically.
A particular case of such an infinite ranged integrand
arises in the Fourier transformation of the linearly screened
11(k)e-1(k). The long range oscillations in this function
arise from the discontinuity at r = Rm of our model potential,
and since this discontinuity is not removed by screening, it
is essential to reproduce it accurately in the screened
potential. We proceed as follows:
We evaluate U(k) analytically and obtain
.Si / Atl i VI 2 eleiz, ',1 de>"7-,
./(Ai 71-
- 53 -
6:,1) _• (.7 C Z
(3.20)
e(k) is calculated from (3.4), so that U(k)e (k) is known.
The integral which gives the Fourier transform of U(k)c 1
is spit into two. For 0 < k < KM, the integral is performed -
numerically.. KM is chosen so that
(3.21)
and consequently the contribution to the integral for k > KM
is approximately
) 47•' 1 ■3 /t1 s-12 : n,tf(vItf 741
3.22)
The function Si'(x) is related to the sine integral function
-eL72.7cA4
S (X) = ( 3.23) e
by
= (7 /2- (9 (3.24)
and therefore has a discontinuity of 7 at x = 0. Thus (3.22)
has a discontinuity of
- 54 -
/ 4 /10 (3.23)
exactly like the unscreened potential U(r). For the pur-
poses of these calculations si(x) was calculated from
(3.26)
for small x, and from the asymptotic from
(X) (3.27)
for large <:(29)
Next, we make some remarks about the numerical Fourier
transforms. if we have a function f(r) tabulated at a set
ofpoints{r.}at constant intervals h, the simplest procedure
to evaluate the Fourier transform f(k) involves using the
trapezium rule for numerical integration. However, this
technique gives rise to components f(k) which are finite for
all k such that
kh = 2117 n = 1,2 1...
Thus, oscillations in f(k) for large k are produced by this
technique. Such oscillations cannot be completely removed
by any numerical technique, but we have adopted a procedure
which reduces these oscillations to an acceptable magnitude.
The function f(r) is assumed to vary linearly with r within
each step of length h and the contribution to Lhe Fourier
transform from each of these steps is evaluated analytica
- 55 -
These contributions are then summed or the computer.
Last -iv, ,*e describe. some of the difficulties involved
in asset.b1:1 conglomeration of numerical anal:sis.
These - round choosing appropriate ranges end intervals
over which 'n evaluate the intermediate functions ,- ppearile
in Fig. 3.1. Since these functions are usually Entegrated
in three dimensions to obtain further functions, their long • •
range parts are heavily weighted. Considerable care has Lo
be taken that spurious oscillations arising from the numerical
Fourier transforms do not produce appreciable effects after
subsequent integration: appropriate tabulations of all the
functions used are vital,
§3.3 Numerical Behaviour
We begin by remarking on the form of some of the func-
tions we have calculated in q-space.
As we have already mentioned, p/(q) contains a term
representing a bound state and a term representing repelled
, band electrons, whilst p2" (q) represents screening electrons •
in linear screening theory. These function typically take
the form shown in. Fig. 3.2. As U is made stronger, p2"(q = 0)
is increased, whilst the repelled charge term in p l(q = 0) is
also augmented. Consequently, the difference pl(q = 0) -
p2"(q = 0), which is always negative in our model calculations,
becomes most negative for the strongest impurity potentials.
For larger q, the difference p7 p,7" becomes positive
As U is increased and <rlp made more shori:.'ranged, he main
- 56 -
effect is an increase in the .range of pl(q) leading to an
increased positive • region :in P 1 - -P 2". This positive region
gives rise to enhanced Charge density on the impurity
(pnt
(r = 0) positive) whilst the negative region gives rise
to transfer of ol::arge out of the impurity cell. Broadly
speaking, we find that the enhancement and transfer have the
following trends at U and rs
are varied:
Enhancement Transfer
Increasing Li increase increase
Increasing rs decrease decrease
Examples of these trends may be seen in the results of
Chapter 4.
The potential Viu which we calculate from the charge
densities is cut off at large q by the function (l-G(q)).
Consequently, the positive region of pl p2" gives rise only
to a very weak potential. This fact is reflected in real
space where Vnt
is attractive throughout the impurity cell
in the systems we have studied.
The computational scheme defined by (3.6) and (3.7)
is found to give most rapid convergence: about 5 iterations
are typically required to establish Vs
accurately. The
scheme (3.6) and (3.8) is less satisfactory because Vs is
closer to U/e than to U.
We are now in a position to understand the behaviour
of the functions Vnt
and pnt
as the iterative calculation nro
ceeds. Typical behaviour is given in Fig. 3.3. As the V
0.0
A : di enhancement
S9
B : ti ), transj-er
- 57 -
2
0
r---1 0.6 ta--> ' 0.4
O.2
o.(.)
-o-2
I 2 (a')
Figure 3.2 Some characteristic charge densities,
axtracted from computations, ta:r.en to
self-consisency, on the system. Na P.)
- 58 -
, L.
.021-
.0
r(a8 )
m r)
r-
-0.04
-0.06
-0.09
-0:10
-0/12
-0.14
-0.16 -
-0.18
1 2 4 3
vni fry d )
system: No. P
----__--
Fiu_re 1-3elf-,7m3:ictcnoy of
('■-tt
Cr) Iyint
(-",,,C), (3.7),
3,2.5„,„ rezuli:s of
- 59 -
becomes mor: attaactive, both density enhancement on the
impurity and ,..fer of charge from the impurity cell become
stronger. I'aecuaa. of the effect of G(q), the charge transi'er-
has the greater effect on the screened potential. Thus V
becomes more attractive still. We see that after 5 iterations,
complete self-consistency has not been obtained. However,
the iterative behaviour is then sufficiently well established
for extrapolation to be reliable.
The self-consistent scheme has been applied to the
systems listed in Table 3.1. This list has been compiled
so as to include systems which should be modelled well by
the scheme (e.g. Mg Si) as well as systems 1 1 WA-; CA will indicate
the behaviour of the scheme as the model parameters are varied,
We are now in a position to confirm our assertion in
§2.2 that 1 2 is not a weak potential for band electrons.
We compare Vs(kv) as computed in our schema with <kr 1 7 )11_,
(where is just Vs) in Table 3.2. It follows from these
numbers, that while Li is weaker than Vs for Fermi surface
electrons, it cannot be regarded as negligibly small.
Table 3.1
Model parameters of the systems studied.
Host Impurity Z1-Z° Ao rs
P 4 -5.67 2 3.93
AP, 2 -2.57 2 3.93
At 1 -1.41 2 z.65
S.-L 2 -2,84 2 2.65
Na
Na
Mg
Mg
60 -
Sys t em
Table 3.2
I Vs (k.„)
Na P 280 106
Na AZ 141 64
Mg A 45 11
Mg Si 89 46
re.
- 61 -
- CHATTER 4
RESULTS AND TTURTERDEVELOPMENTS
§4.1 .Screen -Ln2 Cha
The screening charge densitie,, computed in the prescnt
scheme for the systems listed in Table 7 are given in
Fig. 4.1,
The form of plir(r) first requires some comment.
Since our model U(q) (3.20) has a shorter range than the
Coulomb po*- ential with the corresponding large r behaviouT,
(0 has a longer range than the cnarg,- density induced P 2in
by the corresponding Coulomb potential. No attempt has been
made to compute the long range Friedel oscillations in ozi_1(r);
instead we have concentrated on the charge density within the
Wigner-Seitz cell of the impurity. Even with these restric-
tions,however,typically9970ofthechargecontainedpiin(r )
is present in the region r < 1.5 rs.
For each system, pnL(r) is positive at the impurity
and negative immediately outside the impurity core, The
ratio pni(r=0)/0tin(r=,0) is greatest for the largest values
of Zi - Z°, and at constant Zi - Z is greater for the system
with larger rs . The negative region of p(r) is most sub--
stantial for large Zi
Zo and small rs.
We also compare the total screening charge density
pkin
pnk
with no, the charge density of an electron in the
bound state of the sell'- consiste:ltiv screcved ooten:.ial Vs
- 52 -
1
Figure '4.1 (a)-(d) Electron densitie in linear ar10, non-linear careening th:,.7y,
Curve L
Curve NL p, +
Curve 0 : 7Zo
NIL -3
.03
.02
L •■•■•
•••■•
ti
.01
1 2
(b) Na AC
r(an)
L ea.a. ay. •■■••• MIND IIIN•ir •••■••
+4,
L 1 2_
NL (d) MQ SL
4 r(a)
.06
0b:-
.02.
- 66 -
nt does not exceed 2n within even Within the core region, Pkin + P
in the extreme nynotheticaI case of the system Na P.
Furthermore, Y:'.-1(2 ranee of no
corresponds approrcimately to that
• of that of t.AL short range peak in ozin P.11z . Thus, the
numerical r. ±i.ts are in accord with our hypothesis of a
doubly-degenerate bound state making a substantial contri-
bution to the screening charge density, and our discovery
that this bound state has the effect of repelling electrons
which are not bound.
In the system Mg Ak, which has the weakest impurity
potential of those we have studied, the non-lineer correction
on9.,
is weakest. This prompts us to ask whether a system
which has no bound stage will have an appreciable non-linear
correction to the charge density. A negative answer seems
likely, but in §4.5 a method of calculation, similar to the
present one, is proposed for such a system.
Another vital point in the interpretation of plp(r) is
to recall that it is calculated from a pseudopotential, Con-
sequently, the wavefunction ‹rlip> which enters our discussion
has none of the nodes which the actual bound state wavefunctien
(say, a 4s state) would have in an impurity system. - 119,
therefore lacks structure in the core region. Such structure
may have an appreciable effect in a real system, probably
giving rise to less charge in the core region than our model
predicts.
The overall structure of our non-linear charge density
is similar to that obtained in the non-linear theory of
- 67 -
Sjtilander and Stott(14) in that pnt
represents an accumulation
of screening charge on the impurity. However, unlike their
scheme, which behaves very strangely when applied to attractive
impurity potentials, our scheme behaves in a comprehensible
way even for the extremely attractive potential of P in 1.
54.2. Charge Transfer
The problem of charge transfer in alloys is one of wide
interest. We have therefore presented the effect of our non-
linear correction on charge transfer in Table 4.1, where the
number Q
electrons outside the Wigner-Seitz sphere of the
impurity is given for each of the systems we have studied,
both with and without the non-linear correction pnt.
These
numbers have been obtained by integrating piin(r) and pliz(r)
outwards from the origin.
Table 4.1
Charge Transfers Q in the linear and non-linear theories,
System
alinear anon-linear
Na P
Na A2
Mg At
Mg Si
1.19
0.66
0.25
0.52
. - As we noted in §4.1, our model potential gives rise to
a long range in piti . (r) and so Q is large even in linear
screening theory. The non-linear correction leads to a still
larger Q, the increase being most r:.arked for large Zi
Zo
- 68
and small rs However, these results are critically depen-
dent on the fine balance between the positive and negative
regions CF (r) , so that one cannot put too much reliance
on .the 7r cise numerical values of Q. Nevertheless,
Q does indicate an important physical effect- the non-Ii roar -
repulsicn of band electrons from an impurity arising from
the formation of a bound state.
We have here a partial resolution of paradox. A
theory of alloys which is based on linearly-screened ionic
potentials leads to very small charge transfers(13)
, whereas
a theory which constructs the alloy from renormalized atomic
states(3) predicts substantially larger charge transfers.
The latter theory seems to be confirmed by experiment. We
suggest that the discrepancy is due to the failure of linear
screening theory to treat the poles in the impurity-electron
interaction, and the consequent inadequaey of a description
of screening in terms of a simple RPA screening length.
§4.3 Screened Potentials and Bound States
The screened potentials and binding energies of the
deepest s-states in these potentials are given in Fig. 4.2
for each of the systems in Table 3.1.
Since pnk only represents a redistribution of charge.
Vnk
is weak.nk is attractive throughout the impurity cell,
and this ,is consistent with the transfer of charge out of the
impurity cell represented by pug We have already mentioned
that the absence of structure in V .(r) for small r is due to
- 69 -
the cut-off in k-space imposed by (1-C(k)). This effect
should be most marked for 'small h,, but because of the great
rangesoftheesizeofthestructureino,(k), it has not ru1
• proved pc:-.eible to confirm this trend.
n
11 the systems we have studied V119
is euch we"'_:: r
than Vs
throughout the core region of the impuritycell.
Consequently, provided the binding energy of the bound state
in the linearly screened potential U/e is sufficiently large,
this binding energy is not appreciably modified by V . 119, If,
however, the binding energy in U/e is small, the effect cm
the binding energy of V Ilk may be appreciable.
An important conclusion to be drawn frem the present
work is that, in metallic alloys, bound states, split off
from the valence band, are likely. This is in accord with
the self'-consistent Hartree computations of Sjnander and
Stott(14) who find bound states associated with a single ;,
charge embedded in an electron gas with r, > 2. However,
the presence of states split off from the valence band is
difficult to establish by experiment, Such states are often
far removed from the Fermi energy and have little effect on
transport and thermodynamic measurements. Only in optical
experiments will they play a significant role, soft-X-ray
emission spectra, for instance(3°).
Soft-X-ray emission spectra arise from electronic tran-
sitions from valence states to core levels. In alloys, the
transitions to core states on different types of site may be
distinguished. The intensity ,)f. the spc,r:Zrum reflects the
- 70 -
I .
I ‹Ni (NO 1
Figure 4.2 ()-(d) Screened iripurity DotenTials.
Dashed curve : U lito, screeninr; Vseory).
Sclideur-ve: (non-liner scree:ling), --e to and E, are the energies of ne ':leerest s-states in these two potentials revely.
NO.
•■•• .••• ■•■••
,■••••
I I
viv o 3
.o -
(Pk9
r- (0)
• c9 712 og
7
7:0 --
r , 1 F i 7 ;
-
- 1
8 .0 --
'..""-....... 'a,....,
-
,. ,.., ... • I....
1 ,
S. '..4 .ft
... .
-In-
5'1 -
0 -
pk)
0 -I
- 74 -
local density --.T.71te-nce states of the system at a particular
type of siL- c-. by 'factors such as matrix elements
coupling cor7 112nre states. If a localized state
below the ILInd is present in a system, one therefore
expects a -- the spectrum below ch peak due to the
valence band itself.
Experiments on dilute alloys are difficult because
the solvent spectrum is modified little by the solute, whereas
the spectrum on solute sites is very. weak. Experiments on
concentrated alloys have been performed, and in these there
is evidence of bound states below the valence band: the
spectrum on the sites of higher valence has a split-off peak.
An example is given in Fig. 4.3.
To interpret such spectra, we make the following assump-
tions:
(1) The peak at the lower end of the solute spectrum is
due to a bound state, the broadening being due to
matrix element effects etc.
(2) The upper part of the spectrum is due to the free-
electron band of the solvent valence electrons.
(3) The top of the spectrum corresponds to the Fermi
level.
We can now estimate the binding energies of the supposed bound
states. For a Mg Ai alloy(31)
we obtain En
- 0.5ev and,
for Mg Si(32)
, Eo
3ev. These numbers are no.t inconsistent
with our self-consistent calculations, since non-linear
screening is likely to be modified. In concentrated alloys.
Ai(
171 /
1,1 r' ..1.1■■•■■
- 75 -
4S 50 60
F (e .17 )
Fir;ure 4.3 L23 -emi3sion. spectra on .ziag-liesium and. aluninlurn in the pure Letals (solid c-arve) and in the alloy Aim YE:I? (daFJhed curve) 0 (Data taken. from reference 31.)
76 -
Perhaps better accord with experiment might be ob-
(33) tamed via Jacobs's calculations of soft-X-ray spectra
of Ak alloys, which are in good agreement with the spectra
of.Appleton and Curry(31)
. Jacobs models the alloys by
arrays of square wells of different depths, so that by
choosing appropriate well depths on the basis of our cal-
culations, it should be possible to investigate the effect
on soft-X-ray spectra of our non-linear screening corrections.
rinally, we remark that it may be possible to infer the
presence of corrections such as Vnk
via calculations involving,
the use of screened pseudopotentials, calculations of cohesive
energies, for example. However, all experimental checks of
the present theory rely on the precise form of the pseudo-
potential. so a more sophisticated choice of pseudopotential
than that employed here would be necessary.
§4.4 Diagrammatic Arguments
In the theory we have described, we have used pertur-
bation theory to calculate charge densities and then competed
the solution to the problem by a self-consistent field argument.
It is instructive to see which of the terms of the correct
many-body perturbation series this procedure generates. This
will give us some idea of the deficiencies of our theory and
suggest some remedies.
We make two initial assumptions about our scheme i'3.5).
1. That the treatment of electron correlations in the
absence of the impurity is exact.
- 77 -
2. That our approximate t-matrix, calculated using U1
- is also exact.
(3.6) and (3.7) then generate the diagrammatic terms
• shown in 4.4 which uses the notation of Chapter 1.
We call this series for Vs, Series A. The complete many-
body perturbation theory treatment which was introduced in
Chapter 1 gives a well-known series for V5 which we call
Series B.
We begin by noting that B includes diagrams which
do not appear in A. For example, A only includes "tree- .
diagrams", that is diagrams which can be split up by cutting
a single renormalized interaction y. Thus (a) of Fig. 4.5
is excluded. Also excluded from A are diagrams (b) and (c)
of Fig. 4.5. We may regard. all these excluded diagrams as
being corrections due to exchange and correlation of the
series shown in Fig. 4.6. Since the latter series represents
the contribution of a bound state to a screening - charge den-
sity, we conclude that the excluded terms are a consequence
of neglecting exchange and correlation between bound electrons.
In principle, this could be remedied by using a more sophisti-
cated separable potential (2.10) in which the positions of
the poles were determined by a treatment which included ex-
change (the atomic Hartree-Fock method, for example). In
the light of the remarks following (2,10), this would be very
.difficult in practice.
The next point to note is that Series A does not in-
clude any diagram of Series B more than once, i.e. Series A
proper Verte:r ,ocer4z6 is
PlgILre 4,4 Generation or S rieu for V.
\/2
GM% 01.1ft
1AdIere.
-1( is 1,.t
pf•oree' poletriet.4i.41,7 prze
- 78 -
- 9 -
(a)
(6) (c) (J)
Figure 4.5 Some terzla of Se:ries 13 which Serie A
does not reproduce,
Figure 4.6
- 80 -
does not "overcount". This is consequence of Series A only
including "tree diagrams"; because of the iterative way it is
generated.
Finally, we comment on the two assumptions we made at
the start of this discussion. The assumption that y has
been calcUlated exactly is satisfactory when it couples free
electrons since the theory of Singwi et al.(13) is well tested.
However, in Series A, y may couple states of free character
with bound states, a situation which the theory of Singwi
et al. was not set up to handle. The assumption may only
be justified if a more sophisticated scheme, such as (2.58),
is employed. The assumption that the t-matrix of U1 is
exactly the t-matrix of U is also not really justifiable.
This defect of the theory may be remedied by using the more
sophisticated separable potential (2.10), but we have seen
that such a remedy involves considerable difficulty.
§4.5 Other Potentials
Although the theory we have presented so far is appli-
cable only to impurity potentials which have a self-consistent
bound state, it is readily modified to take account, not only
of weaker attractive potentials, tut also of repulsive
potentials. We outline here how this may be accomplished.
Given a weak attractive impurity potential or a repul-
. sive impurity potential U, we define a constant S such that
the potential BU has at least one 'opund state kb(13)> with
energy E0(8).
- 81 -
(4.1)
For a weak attractive potential 8 > 1, whilst for a repulsive
potential
0. The fictitious state (8)> permits a
separation of ez analogous to (2.3).
&to,)> g)/ /& (4.2)
=
U1
and U2 have similar properties to U, and U2
of Chapter 2,
and we treat them in a similar way, using a theory which is
linear in U2, but summing the t-matrix perturbation series
for U1 thus:
‘(7W > Y'() ,7-(Z (4.3)
Using an identity analogous to (2.20), the denominator of
(4.3) reduces to
fi )/&(7:,/w
(4.4)
Since U has no bound state, we expect T (E) to have no pole
- 82 -
for E < 0, this may only be true for certain values
of We a ± s from our previous analysis that
<(f3)1U Go , have a finite imaginary part in the
band region Consequently, Tl(E) has no poles on the
real E axis, 'i;here it may be written
_(/(/ W3)_> e('-KKZ( (4.5)
T(E) is a complex function of E, non-singular for real E
and which may be computed in practice just as Dr(E) and D(E)
of Chapter 2. In place of (2.37) we now have a corresponding
expression with the factor in curly braces replaced by
/t (Eh) f(E4,1 - 77-17 .4vg(4,,,; ) 6 k — Ekfa
The second term of (4.6) can be written
where •
(4.7)
Consequently, the integrals to be performed in order to
obtain p l (q) take exactly the same form as integrals which we
have evaluated in Chapter 3. The numerical complexity of
83 -
such a calculation would be co,,Iparable with that of Chapter
3, but there would be the added complication of confirming
that the results were essentjally 'independent of the precise
value of S chosen.
§4.6 Summary
We have assessed the effects on impurity screening in
metals of the possible formation of bound states. To achieve
this, multiple scattering of electrons from the impurity has
been studied. Simple models of the impurity potential and
the metallc electron gas have been employed, and we have
found that, compared with a corresporkding theory which neglects
- these effects:
1. There is a large enhancement of charge density on the
impurity.
2. The charge transferred out of the impurity Wigner-
Seitz cell is increased.
3. The screened potential is more attractive throughout
the impurity cell.
The effect differs from one system to another only in a quanti-
tative way.
Whilst the computations necessary for the self-consistent
scheme are tedious, the scheme is applicable to more realistic
models and to other systems with little increase in difficulty.
The scheme provides a link between the screened pseudopotential
and atomic-state viewpoints of the electronic structure of
alloys.
- 84. -
CHAPTER 5
ZERO-GAP ST,T=DITTORS
§5.1 Introd,; ion
A hu.JTher of materials which crystallize in the diamond
or zinc-bicnde structures have transport and optical properties
which are best explained by a model of the band structure which
was first proposed by Groves and Paul(34)
for grey tin, a-Sn.
The model, which has since been applied to Te, Hg S, Hg Se,
Cd3
As2
as well as a 'number of ternary alloys(35) , is charac-
terized by having valence and conduction bands which are de-
generate at the Fermi level and in the centre of the Brillouin
zone. The two valence and two conduction hand states at the
zone centre all belong to the same irreducible representation
of_ the group of the wavevector (r8 for diamond and r8 for the
zinc-blende structure). The degeneracy is thus a direct con-
sequence of the crystal symmetry and these materials are
referred to as "symmetry-induced zero-gap semiconductors".
This degeneracy may be lifted only by breaking th, crystal-
line symmetry (by applying a magnetic field or uniaxial stress
for example). Such symmetry breaking is also expected in
the transition to the excitonic insulator phase, first proposed
by Mott(35)
and examined in some detail by Sherrington and.
Kohn(37) with reference to zero--gap semiconductcrs. However,
the conclusions which we reach in Chapter 6 on the effect of
screening on the formation of donor levels in zero-gap semi-
- 85 -
conductors, suggest that the formation of excitors in these
materials is unlikely. Indeed, no symmetry-breaking elec-
tronic transitions have been observed. The exciton problem
iss-a difficult self-consistent one: the symmetry-breaking
lifts the degeneracy, this modifies the dielectric function,
and this inturn has an influence on exciton formation and
the possibility of the excitonic phase. We shall therefore
adopt the ideal zero-gap band structure in our discussions.
The band structure close to r48 degeneracy of a-Sn is
shown schematically in Fig. 5.1. For comparison, the corre-
sponding part of the band structure et a typical diamond
structure semiconductor, Si is given in Fig. 5.2. We see
that the gapless nature of the a-Sn band structure is a con-
sequence of the re-ordering of the F,- and r8 levels in Si.
The fact that the degeneracy in a-Sn lies exactly at the
Fermi level for the pure material is also a consequence of
this re-ordering.
Perhaps the most interesting consequence of the band
structure of zero-gap materials is the distinctive dielectric
and screening behaviour to which it leads, intermediate between
that of a metal and asemiconductor. However, before discussin
this behaviour in more detail, it is necessary to know the
precise form of the band structure as well some properties of
the states in the band. We shall present this information in
. the next section.
However, we first note another interesting property of
these systems in which the band structure: plays an important
role: the formation of impurity states. Such states have
8
been inferred from recent magneto-transmission and magneto-
resistance measurements (38')
In lig Te, for example, acceptor
states with activation energies of 2.2 meV and O. may have
been detected via both types of experiment, but no donor states
appear to be present. Because the acct to states must be
degenerate with band states, we can expect them to have a
finite resonant width. Since the states are easily detected,
the resonant widths must be small, but they have not been
Measured. It is the nature of these impurity states, the
apparent absence of donor States, the activation energies
and resonant widths, and the influence on all of these of
• screening which will concern us in Chapter 6.
E5.2 The Band Structure
In order to perform calculations on the screening
properties of zero-gap semiconduci- ors, le is necessary to
specify some of the band structure and band states in a rather
precise way. We introduce the band structure via the effective-
/ (391 mass formalism of Luttinger and Kann' ' which we shall require
for the study of impurity states in Chapter 6.
The effective-mass formalism is a method of studying
the electronic states in a crystal to which a perturbing
potential U is applied: it is p*articularly appronriate,
therefore, to calculations on screening and impurity states.
- The aim of the method is to discover a representation in terms
of states in,k), where n labels a band and k is a reduced
wavevector, such that two conditions are satisfied for small
- 87 -
Figure 5.1 Sche:uatic•bana structure ofEK-Sn close to tile :Toerzi level EF • and the zone centre.
Figure 5,2 Scheatic band structure of Si close to the Ferz.i level EF and the zone ce-atre,
- 88 -
(a) U -ae - riot couple states in different bands;
(b) the eert Ho of the Hamiltonian describing
the pure .crystal is represented by
z
(5.1)
where k. and k. are Cartesian components of I
. D. . is known as the effective-mass tensor of the system and
is obtained by using the k.p perturbation method to second
order in k.
In order to obtain a representation satisfying (a) and
(b) several approximations are necessary. First, if U(k)
is a Fourier component of U, k is close to the centre of the
Brillouin Zone and K is a reciprocal lattice vector, we require
(5.2)
Equivalently, U must be slowly varying in real space on the
scale of the atomic cell size. This is a good approximation
for an impurity potential except in the impurity cell itself.
For this reason, considerable effort has been expended in
studying the "central cell" corrations(2) which are parti-
cularly important for impurity s-states. However, we shall
see that: in zero-gap semiconductors the spatial extent of
the impurity states is very large, so thai: tl!einfluence of
/ 7,1/2.41 ;
c j j ✓
(5.3)
- 89 -
the impurity - cell is small. The second major approximation
, involves dro7,nin.- : terms of:order a
2 /a
2i, where a is the lattice
spacing and z. Lhe spatial extent. of the state.
. Again this is an excellent approximation for a zero-gap semi-
conductor.
We now write down the matrix
describing the four r8(r8) bands of a crystal of the diamond
(zinc-blende) structure in the presence of spin-orbit inter-
actions. The symmetry of the system implies that Dnn'
the form
where
D(k)- p
2 0 1,)N
0 * 5
-5 0 S R
(5. 4
P 2A k 2 (k: ,4z- Q - ( kr2 -:yz 2 42)
- ) S Zig 133 (&1 4/1) 2z. 4
(5.5)
Thus, the band structure for small k is defined in terms of
just three effective mass parameters A, B and N. In the
absence of impurities, the bend structure E = E(k) is obtained
- 90 -
from the roots of
(5.6)
' which g;.v-,. us
[(I t
/-* I — R (742,f; z / 2 / Z
-7Lez
(5.7)
where C2
3B2 -
13N
2.
Now in a-Sn and other zero--gap semiconductors, the F8
bands are almost exactly parabolic for small k. We therefore
use the parabolic approximation
C 2
(5.8)
in all that follows. We then have the two doubly-degenerate
bands
(5.9)
It is convenient to define the alternative notation in terms
of the effective masses m and m
(5.10)
It should be noted that alternative expressions to (5.4) in
the approximation (5.8) exist in terms of the 4 x 4 angular
3(40,41) momentum matrix for J. = /
and in terms of the 4 x 4 7.
- 91 -
Dirac y-matrices of spinor theory(42)
Whilst these are
elegant and convenient for certain algebraic manipulations,
(5.8) will prove quite adequate for our purposes.
Lastly we note the magnitudes of the effective-mass
parameters in a-Sn:
A
.242 , 7 19-1
7'n /40
7-2.7.27 ‘pc,-; M4 (5.11)
where m is the free-electron mass. These parameters have
been obtained by many workers (43)
from both optical. and trans-
port measurements. Some discrapancies exist between inter-
band and intraband determinations of effective masses in zero-
- gap semiconductors(44) , but the differences are too small to
have any appreciable effect on the conclusions we reach here.
In a-Sn, as in all the zero-gap systems we have mentioned
my
is much greater than mc. In fact, the band structures of
all these systems have a close quantitative similarity with
each other, and all conclusions we reach regarding a-Sn apply
at least qualitatively to the other systems.
§5.3 Dielectric Behaviour
The distinctive band structure of zero-gap semiconductors
leads to many distinctive features in the dielectric function.
These features have been reviewed by Broerman(45). Here we
shall mention some of the more important conclusions, and
present numerical computations of one feature of the dielectric
function which has not previously been published.
- 92 -
Almost all calculations of dielectric functions in
zero-gap semiconductors have used the RPA. This is not a
very good approximation when applied to an electron gas of
very low density (see Chapter 1). In a zero-gap semi-
conductor, since the number of electron_:,. ci_ose to the Fermi
level and available for screening is smo.11, we may expect the
• RPA to be deficient too. Some - study of the corrections to
RPA has been carried (46)
a ried out and it seems that the RPA results
should be enhanced somewhat. However, these calculations
are complicated and not definitive: we shall use the RPA
throughout, assuming it to give the essential beha-iour.
The dielectric function e(a,w) for a many band system
with eigenstates Ink> and eigenenergies Enk
(n labels the band
and k is the reduced wavevector) is given in the RPA(47) by
the simple analogue of (1.6).
47rez , t 7,t
(v)= g/j-r-y,ri '
h •
f(ErafiL-- keliA) (5.12)
In the presence of deflects, (5.12) is modified by the replace-
ment -f
(5.13)
where Tnn, is the scattering lifetime between bands n and
- 93 -
For a zero-gap semiconductor it proves convenient to
split F...(eM into four terms:
Zrecier, (5.14)
r8 Eimerincludes only interband terms (n n') involving the
r8 or r8 states,
Sinter includes the remaining interband
terms and eintra
contains the remainder of E, in particular
the intraband terms (n = n').
Now sintra vanishes at T = 0 in the pure material.
At finite temperatures and in impure samples, E. has the
same type of behaviour as the intraband contribution to the
dielectric function of an ordinary semiconductor. Similarly,
cinter has no distinctive properties, being independent of q
and co for small q and co. The interesting part is s. r8 inter
,f(e C- inter
eC 6;4 --Z)
+ conjugate expression
with v (5.15)
where v and c refer to the valence and conduction bands.
The states Ivk> and Ick> are obtained as eigen,Tectors of the
matrix (5.4), and in the parabolic approximation (5.8) the
equality
rA:=7 // \
2 3 1- 4/ '7 • •
/ -.4 (;:'
(5.16)
/ Z
o) E()
(5.19)
- 94 -
is established(42)
Using this matrix. element, .the static T = 0, RPA
electric function was. computed by Liu and Brust (42)
For
small q, this takes the form
(5.17)
This behaviour is intermediate between that of a typical
semiconductor
E() = . con,e&-zzz-
and of a metal
(5.18) •
In other words, the screening of a potential in a zero-gap
semiconductor is weaker than that in a metal: the singularity
of the Coulomb potential at q = 0 remains when it is screened
by (5.17). The screened Coulomb potential vs(r) then takes
the form
a
(5.20)
where
7.x ) cas x / -7
.Kz-) (5.21)
and Si(x) and Ci(x) are the sine and cosine integral functions.
- 95 -
In a-Sn, for examole. the effect of the function F(x) is to
0 halve the streith of the Coulomb potential at r = 150A(42)
Because of fact, it has been suggested that the function
F(x) has liLtle effect on the problem of impurity binding,
but this is not so for our model calculations on impurity
states in Chapter 6.
The work of Liu and Brust has been extended most notably
by Broerman. He has shown(48) that the parabolic approximation
(5.8) is not really justified, but that little modification of
the result is required because of a fortuitous cancellation of.
the errors in (5.9) and (5.16). In addition, he has shown
that at finite impurity concentrations and finite temperatures
(when the chemical potential is moved away from the 1.8 de-
generacy), the singularity in e(q) (5.17) at q = 0 is removed(49) .
r
temperatures at various doping and compensation levels. This
is achieved by reducing (5.15) to a one dimensional integral.
e.8
(q = 0) is• finite at all finite temperatures, but peaks inter
strongly when the chemical potential falls close to the
degenerate r8 levels.
In order to gain an understanding of the temperature
dependence of screening and the formation of localized states,
a finite temperature calculation of e.8nte for q -74 0 is required, ir
again in the static limit. This calculation has been performed
numerically in order to facilitate the discussion in Chapter 6.
(5.15) is reduced to the two-dimensional integral:
In particular, he has calculated ein8 ter
(q = 0) for finite
r
r
) (5.22)
-- 96 -
where -(1 • 1+ I 4.2 Yiz 7
Y2(/.-7-1 e- 4 ij 24 Yi;zz, (Y2 -7L
c = ??4
2 r 2
Xo
(5 .23)
and 4T) is the chemical potential. The approximate linear
dependence of IT(T) on T which Broerman has established(49)
xD = — (5.24)
has been employed, and the band parameters of a-Sn in the
parabolic approximation have been used. The integration
presents no numerical difficulties because the integrand is
non-singular at finite temperatures. For the region y = y'
where
(5.25)
we have the contribution
- 97 -
4 OV ^t
f / 0 / r 4 ,g_ - 1 i V
..., ..c.. 4. ..), ,... ,,, . — / : — / I -,` 7 7 (5.26)
to C(a).
Thus, in thc region defined by (5.25), the integral may be
performed exactly, thus avoiding the difficulty of the infi-
nite range of the integral over y.
The numerical results appear in Fig. 5.3. We find
that e. 8 (o) differs appreciably from the T = 0 result inter•
/7
hare", ( (5.27)
only for
r
-') • (5.2E)
The q + 0 limit computed by Broerman(49)
is reoroduced in
this calculation.
The effect on e(q) of magnetic fields and uniax ial
stress has been studied extensively by Liu(50): again the
dielectric singularity (5.27) is removed. However, the shifts
in energy levels induced by typical magnetic fields arfe small
by comparison with ktT in a typical experiment. Consequently,
finite temperatures and finite doping levels are likely to be
the dominant effects removing the dielectric singularity in
most real systems.
- 98 -
I J
-t.
Figure 503 Finite .temperature static :Lerband dielectic r •
function 6.(4.7) calculated from (5.22).
The T.0 lint is cbtainod from (5.27), and
the ticks separata thf; re3ions defined 7)y.
(5020.
- 99
We also note that, at finite temperatures, a sinizu-
larity of the form c2 appears in the static dielectric function.
This arises from the ir.traband term eintra
Corrr!sp_,ndinn anomalies exist in he finite-frequency,
zero-momentu-transfer dielectric function. The calculations
of Sherrington and Kohn(51)
have been extended to finite
temperatures by Grynberg and others(52)
, and the results fitted
to infrared reflectance spectra of Hg Te. This gives a
relatively simple confirmation of the calculations. Un-
fortunately, no such simple test of the finite-momentum-trans-,
fer calculations is possible: whilst the eielectric singuiari-
53 ties must affect the mobility of electrons
() , in practice
the interpretation of experiments is complicated by non-
parabolic effects and scattering from defects
- 100 -
CHAPTER
IMPURITY ZERO-GAP SEMICONnUCTORS
§6.1 Introthiction
The discussion of impurity states in zero-gap semi-
conductors is difficult by virtue both of the characteristic
band structure and the consequent chraeteristic screening
behaviour. We begin, therefore, with a qualitative discussion
• which will illuminate our choice of model to describe such
states.
If we introduce into a material with the band strucYlre
of Fig. 5.7 an impurity which is attractive to electrons, we
expect abound donor state to be formed beneath the conduction
band. However, the general_theory of impurity states (56)
tells us that because the density of states of the system is
finite in this region, the donor state will only be a quasi-
localized resonant state with a finite resonant width.
Similar arguments indicate that there are resonant acceptor
states in the presence of appropriate impurities. A schematic
density of states for a zero-gap semiconductor with both donor
and acceptor states is shown in Fig. 6.1.
Supposing the donor states to be approximately Hydrogen-
atom-like orbitals constructed from conduction band states,
then the Bohr radius of the ground ;tate is
c knor fo
- 101 -
Figure 6.1 Schematic 'density of ztatea 21(13) in
a zero-gap zeioonductor doped duel
with both p- aad n-type
- 102 -
where c is
tit - lectric function. For a-Sn, r
5 x 10-4
ems acceptor -5 r 5 x 10-5 y
Now o
the screenia (1111 of a-Sn is about 5 x 10-5
cms. We .there-
fore expect screning to exert some influence, particUlarly
on donor states. We also note that the large spatial extent
of these states (over 109
and 106 atomic cells respectively)
suggests that central cell corrections will not be significant.
The impurity problem in -Sra was first tackled by Liu
and Brust(55) Their theory, whilst treating the band
structure carefully, concentrated on the effects of the cen-
tral cell part of the potential, ignoring the long-range
screening arising from the dielectric singularity completely.
. Gelmont and others
(41) were able to decouple the four coupled
Schr8dinger equations arising from the four-band model of the
impurity problem using a Coulomb potential. Activatir'n
energies E0 and widths r of resonant states were obtained via
a phase-shift analysis of these equations. In their model,
acceptor Eo
depends on both me and my'
arcep&v".
etcrezoeeo O
ce (
792 c 7, zr
(6.2)
whilst no resonant donor states are possible. The importance
of screening was recognised by Gelmont and his co-workers, but
could not be included in their rigorous calculations.
(38) Recentiv, Bastard and others have reported calcu-
donor
lations based on a similar principle to the calculations of
As well as reproducing (6.2), they find
anor
/7??zr
/ 0 AP74.
(6.3)
- 103 -
this chapter, but using a model potential which, whilst being
non-local, is essentially .of very short range. They find
both acceptor and donor states for appropriate strengths of
the impurity potential.
It is our aim in this chapter to steer a middle course
between the theories of Gelmont et al. and Bastard et al.,
employing neither the infinite ranged Coulomb potential nor
a very short ranged potential, but something of intermediate
range, more appropriate to a Coulomb potential screened in
the charactetistic manner of a zero-gap semiconductor. In
this way, we intend to discuss the effects of screening in
a systematic way. The four-band model is used in the para-
bolic approximation and the central cell correction is ignored.
The Coulombic nature of the unscreened potential is sacrificed,
but the separable model potential has many desirable properties.
The model potential is screened in a variety of ways, and che
dependence of resonant state formation on screening is dis-
cussed.
§6.2 The Model. Potential
We saw in Chapter 2 (2.21), how the Dyson equation is
easily solved for electrons responding to a separable poten-
tial. In this chapter, we wish to calculate a density of
- 104 -
states via a ,:r en's function, In order to be able to cal-
culate the functign simply from a Dyson equation, we
employ a mojr:l ;rn"rity potential which is separable, that
is with a i.lo-mc tum-space representation of the form
(6.4)
Bastard et al. (38) have used
(6.5)
where V is a constant and uk is an isotropic function, equal
to one for small k, and vanishing for some large value of k.
We shall -use the more sophisticated choice
(6.6)
where U(k) is the momentum representation of the potential to
be modelled.
The choice (6.6) in (6.4) can in no way be regarded as
an approximation to the potential U. However, if we take the
concrete example of U being a Coulomb potential, then (6.6)
gives a potential which has singularities and an infinite
range in real space, just as U does. Furthermore, there are
many situations where our choice yields results which differ
from those of using U only by a numerical factor of order
unity. For example, the exchange energy per unit volume E ax
of completely uncorrelated electrons interacting via the
Coulomb potential U is
(6.7)
- e2 4
4 -r-3
(6.9)
Here,
- 105 -
If Eex
is c.alculated using the model potential defined in
(6.6), we obtain
—ex (6.8)
Thus, the model potential yields one quarter of the energy
obtained via the true potential and gives the correct dependence
on physical parameters. In §6L.3, we shall see that (6.6)
yields a ground state energy of the Hydrogen atom which is
four times the true energy.
We shall make the choice (6.6) for all potentials U,
but one must bear in mind that energies computed via (6.6)
will only be approximately correct.
§6.3 The Density of States
We begin by writing down the momentum-space Dyson
equation for the one--electron Green's function G(k,k'; E)
in a crystal containing single impurity represented by a one-
electron potential U,,,.
t 1 W,74
- 106 -
where 11o(k) is the effective mass Hamiltonian of the crystal
and
(6.10)
0 otherwise
If Ho
refers to an n-band .system, then all the functions in
(6.9) are n x n matrix functions: 6 is, of course, diagonal)
and Ukkl is a diagonal n x n matrix by virute of the effective
mass representation used in Ho.
We now replace U. 1 , with the model. separable potential
of (6.4) aid (6.6). (69) may then be re-arranged to give
(-,-:(k,E)441
(,E) f-t
-4-Cr . / - —1 (6.11) 4171p
The density of states of the system is easily written
h
= 7!" K.E)
(6.12 )
Here no
and nimp
correspond to the parts Go
and Gimp
of G,
and the trace is taken over the n x n matrix for G.
The hydrogen atom
As a test of our model potential, we now apply (6.12)
• to a free-electron-system in the presence of a Coulomb paten-
- 107 -
tial to obtain an estimate of the ground state energy of •
the hydrogen atom. We have
-E 2/ 2 I / 71) 0 t 2 292
I -47-ez ) /72 IA/ i?
(6.13)
Now we expect n. (E) to peak at the..binding energy Eo of imp
the system, and (6.12) tells us that such a peak will occur
where (6.11) indicates that Gimp(E) has a pole in the complex
E plane. Thus Eo is given by the solution to
= (;) (6.14)
The sum in (6.14) is replaced by a principal part integral
and
- 2 721.7_ (6.15)
is obtained. This is exactly four times the ground state
energy of the hydrogen atom as we anticipated in §6.2.
Zero-gap semiconductor
We now apply (6.11) to the case where H is the efec- ..;
tive mass Hamiltonian of a zero-gap semiconductor in the
parabolic approximation, Wk being as yet unspecified. We
begin by inverting the matrix E - Ho(k) to give
1
E-±P- .-1. Q 0 S -
0 E--.P-246 -5" R* o /? E)=
--) [E- At3)kZIE- 4-3)k1 R - 5 F - - f P 0
5* R 0 E-4P
=
- 108 -
Next, we examine the elements of the matrix
(6.17)
and consider, for example,
•.(•- 7L)
(6.18)
We replace the sum in (6.18) by an integral and utilise the
fact that Wk is a function of k only. The angular part of
• the integral is readily performed and we prove that (6.18) -
is equal t,
// r 2-/ Lf ----(4/6),,e 2 -5-(4-0k2IF-( ei--,e )kj (6.19)
The other elements of (6.17) can also be simplified in this
way: the diagonal terms are all equal and the remaining
elements vanish.
We also have
4 pi.-- 41e) 2
(6.20)
Thus, (6.19) and (6.20) permit the following explicit ex-
pression for no and n. to be written: imp
_ , 4a` SIP0 /712-12
-714-8)/#2 - (6.21
(6.24)
- 109 -
, (7E-- Zr7Z. 2
14 1/1 1-7 t- /1--- -1/4- ' _1 7R) -1
(6.22)
Density of band states
We now confirm that (6.21) generates the density of
band states of a zero-gap semiconductor. We apply the
identity
--A k" 2 —
Je2 Ir - • // 9 7
to give
)
) (47'/ )A2 ) •
(6.23)
We first consider the case E > 0. Since (A-B) < 0, A+B > 0
and k > 0, it is possible to make the replacements
SYE q/3 „VE( 48))
c(t-
(6.26) K2_
I -7 I—
- 110 -
in (6.23). Therefore,
(6.25)
Similarly for E < 0, we have
(6.25) and (6.26) are just the charge densities we expect
for doubly degenerate bands with effective masses me and mv.
Density of impurity states
The density of impurity states n. (E) cannot be Imp
directly elialuated from (6.22) since the second order poles
in the complex E plane which appear in the summand, lead to
an apparently divergent result. However, if we write
4k 2 12-.e...S1
•--) • ‘.
(6.27)
where X(E,d) and Y(E,d) are both real, we can confirm by
simple differentiation that
z 77i/ri/C1 77- /,12 ‘-// (6.28)
where X(E) and Y(E) are the limits as 0 of X(E,6) and
Y(E,S). For any complex number Z
-7 ‘, 2. C.47,
/ 7 airy Z 74- 72 //-- (6.29)
where n is an integer. (6.28) therefore becomes
, 00(.. - ;
77 (/-X)1 /• Y 2
(6.30)
where primes denote differentiaticn with respect to E.
By comparison with the free-electron result (6.14),
we expect n. (E) to.peak strongly around E = o where imp
( 6 .31)
We therefore expand X(E) about E = Eo
J 9 ) ;
(6.32)
and define
.(6.33)
Then, provided E - Eo is sufficiently small to use only Lhe
first two terms on the right hand side of (6.32), to put
X(E) = 1 in the numerator of (6.30) and that r is approximate
constant over this restricted range of E, we get
(6.34)
- J t-em, L /-)
(6.36)
- 112 -
Thus, is,subject to these restrictions, a
Lorentzian of width NEo). The fulfillment of these
restrictions, ::-cc-ether with. the condition,
r7 ) (6.35)
will be taken in what follows as evidence for the existence
of a well defined, quasi-localized resonant impurity state.
IEo will then be interpreted as the activation energy of
that state, and r(E0) its resonant width.
We note that
so that n. (E) represents a doubly-degenerate impurity state. imp
g6.4 Impurity Levels: Activation Energies and Widths
6.4.1 Coulomb Potential
We now examine the density of impurity states corre-
sponding to the model impurity potential (6.6) arising from
different types of screened potential. We begin with the
case where the static dielectric function is independent of
wavevector. The potential is then
\14 /2 = SZE, k2 /
(
-- 4 7ri) 7-• --(25ok 2
for a singly-charged acceptor
for a singly-charged donor
(6.37)
where co is the dielectric constant.
cc:0
- 113 -
or a singly-charged acceptor impurity, we begin by
calculating H(E) in the range E > 0. Replacing the sum in
the defini':icn of X by an integral gives
e .zF E(3 A)
(6.38)
For E < 0
— X C.) 5 ,
°
(6.39)
For a donor impurity, the sign of X(E) is reversed, and the
solutions to (6.31) are obtained:
6,4
4e02 i3-"A • (acceptor) (6.1:0a)
e4 /
(donor) (6.40b)
These are exactly the ground state energies of hydrogen-
like systems whose effective masses are those of the hole and
electron in our zero-gap material. We note, however, that
as a result of using the separable model (6.6), the acceptor
binding energy is independent of me whilst the donor binding
energy in independent of mv. In the light of reference(41)
this is an unrealistic feature of the model, but when screened
- 114 -
potentials are modelled, both binding energies do depend on
.both m and
To calculate r(Eo) from (6.33), we first note that,
from (6.38)
r (donor or acceptor)
(6.41)
To calculate Y(E), the same technique as was used to derive
(6.25) is employed.
2 _e (acceptor) -17-
_ a ez 77" 4 [E 1 71— /3")
7" ( donor)
(6.42)
Then (6.33) gives
r //z
r(e; 7LA
(acceptor)
(donor)
Since all zero-gap semiconductors have 134-A >> B--A, (6.3+) is
not a good approximation to the density of impurity states
associated with a donor impurity over any meaningful range
of energies. We do not expect well defined resonant done:
• 1
Af'reAft
- 115 -
states in any zero-gap semiconductor, no matter how strong
the impurity potential. 'Resonant acceptor states. on the
other hand, are well defined. Thus our conclusion parallels
(41) that of Gelmont et al. , even though the energies (6.40)
differ from theirs, and the power law in (6.43) differs from
theirs in,(6.2),
Numerical estimates are bound to be rough in the
present model. For reference, we obtain ,acceptor 0
6roeir
_acceptor , 0.2. and r/h
o
6.4.2 Zero-Gap Screening
We now examine the effect on these model calculatic .r
of the introduction of the dielectric function (5.17) applic-
able at small lki to a pure zero-gap semiconductor at T = 0,
For an acceptor impurity, (6.38) becomes
y 2 /
0 . A_F e
776, J 4)k,z-E" 6. —TT; (,'-:. 4 ) -(4,n r- ' I
/ 0-AJAI-F irn-A)A 2J1(
(6.31) cannot now be solved analytically, except for very small
A: what we choose to do is make a numerical solution s valid
• for all A, but restricted to the typical situation
.13-74- A (6.45)
- 116 -
We can then re (6.31) as
n ) (x, (6.46)
Here
-/-0 /( / t±.0
and Eo is the X -) 0 limit of Eo.
We note the limit
4,27, f( )( ) .rz (6.47)
(6.46) is salved graphically in Fig. 6.2. Since f(x)2
approaches zero for large x more slowly than /x' solutions
exist for all r and hence for all X. However, as r becomes
large, the ratio E o /Eo becomes very small. This effect is
illustrated in Table 6.1.
Table 6.1
Reduction of Eo due to screening. E0, Eo and r are defined
in the text.
r E0/E0
10-2 0.55
2.10-2 0.44
5.10-2 0.34
10-1 0.26
2.10-I 0.18
5.10-1 0.069 1 0.040
2 0.025
117 -
For a-bn, taking the approximate values - 6 met
and X - 5 2: cm-1
, we obtain E /E - 0.25: the reduction, o o
in bindinc; -_,, n,11- gy is substantial.
We no,,, c,.amine the effect of screening on i (E ). We
see from Fig. E.2 that, close to E E0, f(x) is slowly
varying function of x. This permits us to write, for a- n,
close to E = Eo
X(() 0.4S (6.48)
where X is the X 4- 0 limit of X. It follows that X'(E0) is
1.6 times its value in the abs ence of screening. Y(Eo) is
evaluated in the usual. manner and is -0.67 times its value
in the absence of screening. Finally, the width of the state
is reduced by -0.42, giving
/7
0 3 (6.49)
--0
We conclude that well-defined acceptor states are still
possible in the presence of screening, but that the binding
energies are reduced, and the relative widths of the states
are increased somewhat.
We now turn our attention to donor states. The same
analysis as for acceptor states is possible, and the acti-
. vation energies are obtained by solving, subject to (6.45),.
where
Vy)]2
9,7y) > 0 (6,50)
= 0,169/V4
- 118 -
This is carried out in Fig. 8.3. Now, however, g(y) becomes
negative for y > 0.2, and no solutions are possible for
s > 0.23. Using the parameters of a-Sn which were employed
earlier, we obtain s - 10. Consequently, even taking account
of the considerable quantitative defficiencies of the model
potential, it seems certain that the screening in pure a-Sn is
sufficient to prevent the formation of even weak donor resonances.
6.4.3 Metallic Screening
In a crystal of a-Sn, dilutely-doped with p-type im-
purities, we have seen that quasi-localised acceptor states
are expected. However, we have neglected effects arising
from more than one impurity. The treatment of the effects
high impurity concentrations is beyond the scope of the present
technique, but one important many-impurity effect can be men-
tioned. This arises from the metallic-like screening which
arises because the Fermi level is shifted from the r8 de-
gene racy.
We therefore study the effects of using the model
potential
i/v/4 2 = 4 ye z
42 7: 4, I Ik (6.51)
To examine the acceptor activation energy E0, we solve the
analogue of (6.44)
2- CJ
6-0 L
(6.52)
7
7 2 f
J
(6.53)
=
- 119 -
In the ap?roximation (6.45), : .52) reduces tr,
where
1 2. )
- (z) 2Fc)
-- z -z I' //, j
As a consequence of the limit
Ze4:72 ji (6.5)
no solutions to .(6.53) exist for large t. From Fig. 6.4 we
obtain the limiting value t =, 0.48, and we conclude that crit •
no quasi-bound acceptor states may be formed if the screening
parameter K is greater than about 106
cm-1
. The dependence
of Eoo upon t is given in Table 6.2.
- 120 -
Table 6.2
Reduction.. of -hilding energies due to metallic screening.
Eo, and t are defined in the text.
E /E
5 1
- i0 2
-9 2.10 -
O.70
0.56
O.45
O.35
O.25
O.14
O.00
2.10-1
4.8.10-1
Now in the Fermi-Thomas approximation, K is related
to the number density no of charge carriers and the Fermi
energy EF by
K 2. = Z7-77-6, e 2-
(6.55)
Consequently the critical concentration is reached at a con-
centration of about 10 holes cm-3
or 1016
electrons cm-3
in a-Sn. However, since
(6.56)
we cannot place too much reliance on these critical cancan-
trations.
At low temperatures, we 1:now that holes are quasi-
localised in acceptor states, so that the total hole concel-
- 121 -
. .
Figure 6.2 Graphical Solution of Equation (6,46).
The LHS and RHS are plotted asainst x, the
former for several values of r. For each
the solution is that value of x• at which
LHS and•RHS intersect,
- 122 -
Figure 6.3 Graphical Solution. of Equation (6.50).
The LES and RES are plotted against y, the
former for the 1212..x1mum value of s for which
a solution exists. That solution is thp
value of y at whicli TiLIS and RHS touch,
- 123 -
C5 it
ti 0 d
U)
co 0 0
Figure 6.4 Graphical Solution of Equation (6.53).
The LHS and RHS are plotted against z, the
former for several values of t. For each t,
the solution is that value of z at which
LHS and RHS intersect.
- 124 -
tration should not be used in .(6.55). We may nevertheless
be sure tIlat at :u.fficientiy high concentrations of acceptors,
the concenration of holes of a free character will be suf-
ficiently hig,1 to prevent (self-consistently) quasi-locali-
zation of accc-ptur states.
n-type impurities are not expected to have bound im-
purity states at low temperatures. Consequently, impurity
electrons are freely available for screening. This suggests
that acceptor localization is less likely in compensated
samples than in samples containing p-type impurities only.
§6.5 Conclusions
We begin by discussing the effects of temperature on
the calculations of this chapter. We have seen the di-
electric singularity used in §6.4.2 is removed at finite
temperatures, but that departures from singular behaviour are
important only for
4 7-- 2 2 Q Z
27.7 1/. 712e /Y4, /
Consequently, at temperatures (kB T < E ) where the effects of - o
bound states are Most easily detected, the finite temperature
modification becomes appreciable only for
---Z 172v7
(6.57)
The small range of q over which this• applies has only a small
effect on the integrals X(E) and Y(E) and our concluSions are
- 125 -
not significantly modified.
By far the most important effect of finite temperatures
is the introduction of a singularity in the intraband part
of the dielectric function, and the consequent metallic
screening as in §6.4.3. A temperature of 1K is sufficient
to introduce the same degree of screening as about i016
conduction band electrons per cm3. We may therefore expect
a dependence on temperature of the acceptor activation energy.
Such dependence has not been observed, presumably because of
the low temperatures at which acceptor activation energies
have been iaeasured.
Finally we summarise the results contained in this
chapter. The presence or absence of quasi-localized impurity
states in a zero-gap semiconductor is found to be critically
dependent on the type of screening in the system and the
masses of the touching r8(r8) bands. Acceptor states are
expected in dilutely doped samples at low temperatures, whilst
donor states will be absent. The metallic screening at
finite temperatures and at higher doping levels (especially
in compensated samples) may inhibit the presence of even
quasi-localized acceptor states.
- 126 -
APPENDIX - UNITS
In accrdance with standard usege, two types of units
are used ir tlis thesis.
In Chapt ,, rs 1 - 4, atomic units are used with energy
measured is rydbergs, and length measured in terms of aB'
the Bohr radius. This system is produced by putting the
fundamental physical constants
me = 47re o =
-- =c=e=1
Chapters 5 and 6 employ the c.g.s. system.
- 1 7 -
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